Pension Fund Systems

ABSTRACT

There is provided a computer-implemented method of estimating a capital reserve requirement to cover the longevity risk exposure of a financial instrument in the case of a future longevity shock, the financial instrument undertaking to pay to an investor sums according to a payment schedule of amounts arranged to match with the future cash flow obligations of a pension scheme to at least a portion of its members. The method comprises: (a) calculating, using computing apparatus, an expected payment schedule of the financial instrument by calculating what the cash flow obligations of the pension scheme to its relevant members would be in the case of an expected longevity scenario for the pension scheme membership occurring; (b) calculating, using computing apparatus, a present value of the financial instrument in the case of a stressed longevity scenario for the pension scheme membership in which a longevity-related shock to the expected longevity scenario of the pension scheme membership occurs; and (c) calculating, using computing apparatus and using the calculations of the expected payment schedule and a present value of the financial instrument in the case of a stressed longevity scenario, an estimate of the longevity capital reserve required to ensure that the future cash flow obligations of the financial instrument would be covered in the event that the stressed longevity scenario were to occur.

CROSS REFERENCE TO RELATED APPLICATIONS

The present patent application claims priority under 35 U.S.C. §120 toU.S. patent application Ser. No. 12/117,306, filed on May 8, 2008, andto U.S. patent application Ser. No. 12/212,133, filed on Sep. 17, 2008,the entire contents of both of which are herein incorporated byreference. The present patent application also claims priority under 35U.S.C. §119(a)-(d) to United Kingdom patent application serial nos.0709036.8, filed on May 10, 2007; 0716979.0, filed on Aug. 31, 2007; and0721690.6, filed on Nov. 5, 2007, the entire contents of each of whichare herein incorporated by reference.

FIELD OF THE INVENTION

The present invention relates to the development of a methodology andsystem for securitizing pension liabilities, enabling the introductionof debt capital to achieve risk transfer from the pensions and insuranceindustries. The invention includes the development of a pension riskmanagement system. Various aspects of the invention are also ofrelevance in other environments.

Some aspects of the invention are concerned particularly withimmunization of risk in the pension and insurance sector using, forexample, securities and derivative products to transfer the riskassociated with pension liabilities over to the capital markets. Aspectsof the invention also relate to systems which support the securitizationof pension liabilities, report on the securitization of investments andensure compliance of the securitization scheme with rating agencyrequirements. Further aspects of the invention provide reporting toolsfor corporate sponsors and pension trustees to help ensure theircompliance with regulatory reporting requirements. Amongst other things,aspects of the invention provide methods for defeasing risk associatedwith pension liabilities, systems for supporting such methods, andrelated financial instruments.

BACKGROUND OF THE INVENTION

Demographics throughout the world are pointing to a global pensionscrisis both in the public and private sectors. Mortality improvements,especially at older ages, make it ever more likely that individuals withinadequate pension arrangements will end their lives with insufficientincome and, in some cases, in poverty.

For some private corporations operating defined benefit pensions schemesin which the amount of pension is determined by, for example, the lengthof service and the salary of an employee, the total size of theobligations on a pension scheme sponsored by the employer has grown dueto improvements in mortality. In many cases this has been to an extentthat it has become a significant burden on the corporation's financesand operations and many schemes are operating at a significant deficit.

Concerns to ensure that companies are properly equipped to meet theirpension obligations have seen the introduction over about the last fiveyears of a combination of both accounting and regulatory reforms, whichhave in themselves added to the pensions burden on corporate sponsors ofdefined benefit pension schemes.

Recently adopted international and domestic accounting standards, suchas FRS 17, IAS 19 and FAS 87, now require many companies to reflecttheir pensions deficits on their balance sheets as obligations to thirdparties. Under these accounting standards, pensions liabilities arerequired to be valued by discounting obligations to pensioners on thebasis of long term bond yields, while the assets supporting the scheme,which typically comprise a variety of asset classes in addition tobonds, such as equities and property, are simply recorded at marketvalue. The result is that there is usually an imbalance between thevaluation of the assets and liabilities of a scheme, which can lead tounwelcome volatility in the size of the surplus/deficit. Thissurplus/deficit volatility will ultimately be reflected in the company'sbalance sheet, with the expectation that accounting standards willeventually require this volatility to be included in the profit and lossstatement with a potentially significant impact on earnings.

Further, to date, the development of systems in the pensions sector hasbeen driven by the needs of actuaries and pension consultants, with afocus on the management and reporting requirements of insurancecompanies and pension trustees. At the pensions scheme level, thestandards of record keeping and risk management are generally not of ahigh standard. At the insurance level, the focus has tended to be oncash flow projection and pricing. By capital markets standards, theworld of pension risk management and reporting has mostly beenunsophisticated.

An illustration of the problem is that despite the introduction of theaccounting standard FRS 17, which requires companies to value theirpensions liability on the basis of long term corporate bond yields, itremains the custom to only revalue the liability every three years.Further obfuscation of the true extent of corporate pensions liabilityis provided by the fact that sponsors have not been required to disclosetheir mortality assumptions. This means that despite the move by theaccounting profession to make companies accurately reflect their pensionliabilities in their financial accounts, the reality is that themeasurement has only updated at intervals such as every three years andis then based on discretionary mortality criteria.

Further, recent legislation in some jurisdictions such as the UnitedStates and the United Kingdom requires corporate sponsors to demonstratethat where a deficit exists, they will be able to fully fund the deficitwithin a fixed period. For example, under current legislation in thoseterritories the periods have been set at seven and ten yearsrespectively. In view of this, in the UK a Pensions Regulator has beenestablished with powers to intervene in corporate affairs, including theability to divert dividends or other distributions away fromshareholders to the fund the pension deficit.

Additionally, through quasi government agencies such as the PensionBenefit Guaranty Corporation in the USA and the Pension Protection Fundin the UK governments are being forced to become the underwriters oflast resort of risk of sponsor failure. As a result, in turn theseagencies are now imposing annual levies on the corporate sponsors.

In view of the inadequacies in the frequency and quality of currentpensions reporting, it is difficult for regulatory bodies andgovernmental protection funds to gather accurate or timely informationto enable a meaningful assessment of the ultimate exposure of pensionschemes.

Pension fund problems could clearly cause underperformance on the partof sponsor companies, which could create issues for existingshareholders and potential investors.

Against this increasingly burdensome background, companies are realizingthat the promises made to their pensioners are exposing their businessesto additional and sometimes highly volatile risks, such as inflation,exposure to the interest, currency, credit, equity and property markets,as well as longevity.

In view of the burden of these risks and exposures on the corporatesponsors of defined benefit pension schemes, the management of suchcompanies may choose to close existing schemes to new members, or toreduce benefits and increase the retirement age, or to migrate away fromdefined benefit pension schemes towards defined contribution schemeswhich may not be an attractive alternative for its employees. Thisunnecessarily limits the corporate sponsor as to what is in the bestinterests of its particular employees and business imperatives. However,none of these strategies in themselves will deal with the fundamentalproblem of the exposure of the corporate sponsor to the volatility ofthe deficit, or indeed a surplus which has been the case at varioustimes. Closing the scheme is an inflexible and final solution which doesnot permit the sponsor to claw back a growing surplus, should marketconditions become favorable after closure.

Another option is to abandon the sponsorship of the corporate pensionschemes altogether by transferring the scheme, for example, to anindependently managed collector fund. Such an approach removes theburden of the deficit/surplus volatility, but is strongly discouraged bythe pensions regulator.

Current options taken by companies often have human resourceimplications, with dissatisfaction amongst the workforce and in somecases industrial action as a way of expressing objections to proposedchanges to a company's pension arrangements.

Currently, one source of underwriting capacity for the risk of longevityis the insurance sector, through the issuance of bulk annuity policiesby a multi-line insurer, or a new breed monoline pension “buy-out”company and in turn the re-insurance market. This bulk annuity providesa full legal and economic transfer of the pension scheme's risk bytransferring to the insurer all risks and future liabilities of apension scheme in return for a priced premium and winding-up the scheme.While offering a partial solution, the capacity of the global insurancemarket to assume the risks associated with longevity is extremelylimited in scale when set against the size of the global pensionsmarket, making this an unscaleable solution. There are currently severelimits on the capacity of the insurance sector to supplement itsexisting capacity due to the high cost of capital for participatinginsurers. The high cost of capital arises because participating insurersare required to maintain high levels of regulatory capital largely inthe form of expensive equity capital. This makes a buy-out of a pensionscheme and replacement with a bulk annuity a very expensive andinefficient solution.

A further constraint of the annuity market is that it offers a productbest suited to defeasance and closure of pension funds, rather than asource of risk transfer for existing ongoing pension schemes. The reasonfor this is that pension schemes are not allowed to give preference tospecific scheme members and so bulk annuity is primarily used to defeasethe obligations of an entire scheme.

As an alternative to a full buy-out of a pension scheme, some insurancecompanies are offering to take on schemes' liabilities in a phasedapproach as a partial defeasance of the longevity and other risks. Theaim is that benefits are insured gradually over time allowing the costto be spread and the scheme risks to be managed towards buyout. Somemarket entrants are using this to target small to medium sized companiesand schemes that may not have the available capital for a full buyout.

Another option available to trustees and sponsors of defined benefitcorporate pension schemes is a range of products called pensions riskinsurance. These insure certain risk experience within predeterminedbands over a stated period of time, which may for example be the fundingrecovery period for the pension scheme. For example, this may be tounderwrite mortality and investment experience up to a stated level overthe recovery period.

Ultimately all of these products are categorized as an investment in aninsurance contract. While through a variety of derivatives of the basicbulk annuity product, it is technically possible for a pension scheme to‘invest’ in insurance products as a general asset of the scheme, ratherthan member specific policies, there are significant legal and securityimplications in doing so, as an insurance policy, unlike a bond, is notan unconditional promise to pay, but rather a contingent contract,subject to there being no available defenses. For this reason, insurancederived products, such as bulk annuity are not considered suitableinvestments by many pension trustees and their advisors.

The present inventors have appreciated that investment in bonds, orinterest rate and inflation derivatives can offer a solution to hedgeagainst the exposure of a pension scheme to equity risk, interest raterisk and inflation risk, and would immunize the scheme's liabilitiesfrom ballooning as a result of further falls in bond yields. However, ithas also been appreciated that in many cases this solution would beincomplete as the pension scheme would remain exposed to longevity risk,i.e. the risk that a scheme's pensioners will live much longer thananticipated.

A preferred approach would be to hedge the pension schemes against allof their underlying exposures, including longevity, in order to immunizethem against risk. This longevity risk has thus far been unmanageableand the present inventors have developed systems for transferring thislongevity risk, as well as the other risk exposures and volatilities,away from corporate sponsors and managers of pension liabilities.

The possibility of creating financial instruments which can hedge thespecific economic risk of increasing longevity has been proposedpreviously. There have been proposals to develop and introduce productsin the form of longevity bonds and longevity derivatives which purportto immunize against longevity risk. Mortality bonds, hedging theinversely correlated mortality risk borne by insurers in their lifeinsurance business, i.e. early death, have also been issued.

A longevity bond was announced in November 2004 by BNP Paribas on behalfof the European Investment Bank (EIB). This was proposed as a solutionfor financial institutions looking to hedge their long-term longevityrisks. The bond issue was for £540 million, and was primarily aimed atUK pension funds. The bond was due to pay a coupon that would beproportional to the number of survivors in the cohort of individualsturning sixty-five in the year that the bond was issued, so that thecoupon in each successive year would be proportional to the number inthe cohort that survived each year. Since this payoff would in partmatch the liability of a pension, the bonds would create an effectivehedge against longevity risk.

However, a number of problems with the EIB longevity bond meant that itdid not generate sufficient interest to be launched, and was withdrawnfor potential redesign.

The present inventors have appreciated that a significant inadequacy ofthe EIB bond or any similar proposals for use in the pensions sector,would have been that the mortality of a reference population was used todetermine the payment of the bond coupon. This means that a basis riskfaced by any individual pension plan, namely the mortality circumstancesexperienced by that particular pension plan, would not be covered, thusnot making the bond an effective hedge against an individual pensionscheme's longevity risk.

The present inventors have thus appreciated if longevity bonds orderivatives are to be of use in the pensions sector, they will have toprovide a much more complete hedge for the mortality risks actuallyborne by each individual pension scheme, or at the very least need to beindexed to the mortality experiences of a much greater range of cohorts.

Longevity indices have been proposed, for example by Credit Suisse in2006 by BNP Paribas and most recently by JP Morgan, which introduced anindex under the brand name Lifemetrics, with an aim of creatingbenchmark values for underlying mortality rates or cumulative survivalrates. However, the creation of indices does not move the market anyfurther forward in terms of identifying new capital willing to take onthe risk of longevity, and without this capacity a longevity derivativesmarket is unlikely to take off.

The inventors have identified that a key factor in the growth of thelongevity securitization market is the development of longevity bondsand longevity derivatives capable of hedging the entire economic risk ofan individual pension scheme (i.e. the element of exposure which is leftif an investment or hedging instrument does not exactly mirror thelongevity profile of the pension scheme). The inventors have realizedthat such products would provide buyers and counterparties in the formof individual pension funds and monoline buy-out specialists andmulti-line insurers looking to hedge themselves and their own exposureto the longevity risk, with a complete solution to their risk transferrequirements. Also, the capital elements of such products could createsufficient value to generate buying interest from speculative investorsfor which exposure to longevity products would create an attractivediversification since it is uncorrelated with many of the moretraditional asset classes.

In this regard the inventors have developed a capital marketsmethodology and system for securitizing pension liabilities, enablingthe introduction of debt capital to achieve risk transfer from thepensions and insurance industries onto the capital markets. Theinventors have also developed a pension risk management system tooperate the methodology. This methodology and system were first set outin detail in United States Patent Application Publication No.US-A1-2008/281742, published 13 Nov. 2008, of which this application isa continuation-in-part, and International Patent Application PublicationNo. WO2008/139150, published 20 Nov. 2008.

This capital markets methodology enables immunization of risk in thepension and insurance sector using, for example, securities andderivative products to transfer the risk associated with pensionliabilities (including longevity risk) for a particular pension schememembership over to the capital markets. The associated risk managementsystem supports the securitization of pension liabilities, reports onthe securitization of investments and ensures compliance of thesecuritization scheme with rating agency requirements. The riskmanagement system also provides reporting tools for corporate sponsorsand pension trustees to help ensure their compliance with regulatoryreporting requirements.

This capital markets methodology allows the Trustees of a Pension Schemeto meet its payment obligations over the years whilst reducing the riskof going into deficit.

According to the inventor's methodology, the risk is transferred to acompany which analyses the scheme and its members carefully. The companycalculates nominal cash flow requirements for periods extending over anumber of years. It then calculates the life expectancies of members ofthe pension scheme, using statistical techniques based on lifeexpectancy data for a general population, and factors specific to themembers of the scheme. Once life expectancy data has been calculated,projected actual cash flow requirements are calculated by manipulatingthe nominal cash flow requirements using the life expectancy data. Thecompany, in return for funds provided by the Trustees of the pensionscheme, issues a financial instrument which undertakes to pay sums equalto the projected actual cash flow requirements over the life of thearrangement.

The methodology is able to deal with unexpected changes in factors whichresult in increases in the cash flow requirements beyond those whichhave been projected. Reasons for such changes include rises ininflation/the cost of living so that indexed pensions payments increasemore than expected, and changes in life expectancy. If people live forlonger than estimated originally, then in any particular year, pensionsmust continue to be paid to more people than originally estimated.

This is achieved by providing a financial instrument by which cash flowrequirements will be met despite unexpected changes in such factors bythe financial instrument providing increased or decreased sums to matchthe increased or decreased cash flow requirements, but also protect theissuer of the financial instrument.

The system for recalculating the sums to be paid to the pension schemeto match its cash flows, is as follows. At a re-set point, revisednominal cash flows for each of the original members of the scheme arecalculated taking into account the actual experience of the schememembers in all non-mortality factors affecting pension payments, such ascommutations, transfers out, etc, whereas the actual mortalityexperience of the deaths of any pension scheme members in the precedingperiod are not taken into account in calculating the revised nominalcash flows. That is, if a member has died, the nominal cash flows forthat member remain in the calculations. Actual mortality experience ofthe pension scheme membership is then taken into account by being usedin conjunction with the revised nominal cash flows to calculate anadjusted cash flow for that re-set period.

If mortality experience were taken into account at an individual memberlevel, the nominal cash flow for a deceased member would be taken out ofthe calculation of an adjusted cash flow for that re-set period and forthe calculation of an adjusted cash flow in any subsequent re-setperiods. This approach can be taken in calculating an adjusted cashflow. Instead, in some embodiments of the inventor's methodology,members are allocated into the relevant one of a number of segments,each segment representing a range of nominal pension cash flowrequirements. Within each segment, the revised nominal cash flows forall of the members in that segment are summed, including those fordeceased members, and average mortality rate for that segment is alsocalculated from the cumulative actual mortality experience of thatsegment. The resultant average mortality rate for each segment is usedtogether with the sum of the revised nominal cash flows for that segmentto calculate an adjusted cash flow for that segment. The adjustedsegment cash flows are aggregated to give an adjusted cash flow for thatre-set period which is paid to the Trustees of the pension scheme.

Of course, any party other than the trustees of the pension scheme caninvest in the financial instruments of the inventor's methodology. Inparticular, any party having an exposure to the pension scheme and thefinancial risks associated therewith, including longevity risk, maychoose to invest in a financial instrument provided according to theinventor's methodology. For example, an insurance company underwriting apension scheme may choose to invest in such a financial instrument whichmay transfer any aspect of the risk exposure of the insurance company tothe pension scheme on to the capital markets. Also, any party whoconsiders the financial instrument to be mis-priced may choose to investin a financial instrument according to embodiments of the presentinvention.

This ‘longevity’ financial instrument of the inventor's methodology isnot limited to cash form including bonds, notes, paper, etc., and can bedeployed in the form of a derivatives contract including swaps, options,etc.

The financial instruments can be used to hedge against the longevityrisk and longevity basis risk associated with defined benefit pensionschemes.

The inventors have thus provided methods and systems of securitizing theliabilities of a pension fund to immunize it against its underlying riskexposures, including longevity and longevity basis risk.

The risk management systems are arranged to manage the assets andliabilities of a defined benefit pension scheme and facilitate risktransfer to the capital markets.

This methodology can provide more accurate indications of the risks of apension scheme, in which for example at least longevity calculations arebased on factors associated with the individual members of the scheme,rather than on estimations based on a sample of the general population.

The inventors have thus developed a suite of capital markets basedsecurities and derivatives and proprietary risk management and reportingsystems, which enable multi-faceted risk transfer of longevity and otherrisks from the pensions and insurance sector to fixed income capitalmarket investors.

These financial instruments can be provided as both indexed anddedicated defeasance products, which are capable of assuming the entireeconomic risk of a pension scheme—including longevity (includinglongevity basis risk), inflation, interest rate, credit and equity—bypartially or completely replacing the scheme's existing assets withsenior secured securities or derivatives, which are designed to matchthe obligation of the scheme. That is, the defeasance products arepriced by analyzing the underlying pension scheme's exposures tolongevity risk on a “granular” basis, i.e. on the basis of the pensionscheme's members' actual characteristics, thus allowing more accuratepricing than previously.

This enables corporate sponsors of defined benefit pension schemes toimmunize their obligations from the underlying exposure to risk,including longevity risk and basis risk associated with longevity.

The risk management system provides an operating platform for thesecurities and derivatives. The securities and the derivative productsare capable of being rated by the world's leading debt rating agencies.The senior tranches are preferably be rated highly by an appropriateleading rating agency, for example being rated AAA or Aaa by anindependent ratings agency such as Standard & Poor's or Moody's.

To support this rating of the securities and derivative products, theinventors have also provided a ratings method in which thesecuritization of longevity risk is measured and monitored by the riskmanagement systems to deterministically or stochastically map the actualand projected mortality experience for the pension scheme and allocaterisk capital based on a proprietary risk capital model to ensure dailycompliance with a set of criteria agreed with at least one ratingagency. This permits the securities ratings to be defined, monitored andmaintained.

The risk management system further provides pensions reports toregulators, stakeholders, and pension scheme trustees, enabling theholistic reporting of both the investments and the pension's liabilitieson a daily marked to market basis. This represents a revolution in termsof the business process compared to existing systems, enablingtransparent daily reporting of a pension scheme's assets andliabilities.

SUMMARY OF THE INVENTION

It will be appreciated that there are many different aspects of thepresent invention, and that in the practical implementations of theinvention, one or more aspects may be used together in any operablecombination. Some aspects and particularly the systems may be used inenvironments other than the pensions sector. The accurate estimation oflongevity may be used in a number of scenarios, whether as applied to agroup of people such as the members of a pension fund or to anindividual—for example to determine the probability of that individualattaining a particular age.

In the inventor's methodology, the payment schedules of the longevityfinancial instruments are initially established as being equal to theexpected future pension scheme liabilities calculated on the basis ofthe expected mortality tables for the pension scheme members. Theamounts in the payment schedule of the financial instruments are thenre-set throughout the lifetime of the financial instrument to take intoaccount the actual mortality experience of the pension scheme such thatthe payments ‘mirror’ or match the pension scheme liabilities. Thus thepayments that are actually made on the longevity financial instrumentcan, and typically will, deviate from those of the initially establishedpayment schedule. This can occur, for example, if the actual mortalityexperience of the pension scheme membership turns out to be verydifferent from the expected mortality tables for the pension schememembers that were used initially. This potential deviation of thepayments on the financial instrument away from that initially expectedprovides an unwanted risk source in the financial instrument. When thecause of the deviation is due to actual mortality experience differingfrom that which was initially expected, the risk source is ‘longevityrisk’.

The inventor's methodology attempts to reduce this longevity risk to alow level. In this regard the inventors recognized that standardmortality tables generated for a general reference population may not besuitable for application to the specific members of the particularpension scheme to which the financial instrument is tied. Thisunsuitability of standard mortality tables would, over the course of theduration of the financial instrument, cause the payments that must bemade on the financial instrument to cover the pension scheme liabilitiesfor the specific scheme members to deviate significantly from theinitial payment schedule, as a result of the actual mortality experienceof the pension scheme members turning out to differ significantly fromthat originally projected by the standard mortality tables. Thisuncertainty of how the mortality experience of the pension schememembers will differ from that projected by standard mortality tables ismanifested in the longevity financial instrument as ‘longevity basisrisk’ in the payments that have to be made.

To attempt to reduce the longevity basis risk in the longevity financialinstruments of the invention, the inventors have developed a methodologyof more accurately projecting the expected longevity of the pensionscheme members to which the financial instrument is tied. Thismethodology, set out below, was first set out in United States PatentApplication Publication No. US-A1-2008/281742, published 13 Nov. 2008,and is claimed in United States Patent Application Publication No.US-A1-2009/037258, published 5 Feb. 2009. The inventors have recognizedthat there are a number of more specific risk sources contributing tothe total longevity risk.

One such risk source is ‘trend’ risk. To mitigate this risk source, inthe inventor's methodology, longevity ‘trend’ risk can be reduced byadjusting the standard mortality tables by modeling underlying mortalitytrends in a relevant reference population, typically for which thestandard mortality tables were drawn up, and adjusting the standardmortality tables accordingly. This is intended to minimize the ‘trend’risk associated with the transaction.

Another such risk source is ‘level’ risk. To mitigate this risk source,in the inventor's methodology the standard mortality tables are alsoadjusted by analyzing the socio-economic and other relevantcharacteristics of the pension scheme membership at an individual or‘granular’ level, modeling the likely effect of these characteristics onthe mortality of the individual members, and applying mortality leveladjustments to account for these characteristics to create individualmortality tables for each specific pension scheme member. This isintended to minimize the ‘level’ risk associated with the transaction.

However, even after making these adjustments to reduce longevity risk,there inherently remains some longevity risk exposure in the financialinstrument. This risk needs to be supported by an appropriate amount ofrisk capital. To size the risk capital required to support longevityrisk, the inventors have also developed a methodology of quantifying theremaining exposure of the financial instrument to the various sources oflongevity risk. In this way the longevity risk capital requirement tosupport the financial instrument can be evaluated. The inventors havealso developed a methodology of determining the ‘longevity’ risk capitalrequired to ensure that the financial instrument obtains a given ratingfrom a rating agency, such as, for example, Standard and Poor's, Moody'sand Fitch.

Thus, viewed from one aspect, the present invention provides: acomputer-implemented method of estimating a capital reserve requirementto cover the longevity risk exposure of a financial instrument in thecase of a future longevity shock, the financial instrument undertakingto pay to an investor sums according to a payment schedule of amountsarranged to match with the future cash flow obligations of a pensionscheme to at least a portion of its members; the method comprising: (a)calculating, using computing apparatus, an expected payment schedule ofthe financial instrument by calculating what the cash flow obligationsof the pension scheme to its relevant members would be in the case of anexpected longevity scenario for the pension scheme membership occurring;(b) calculating, using computing apparatus, a present value of thefinancial instrument in the case of a stressed longevity scenario forthe pension scheme membership in which a longevity-related shock to theexpected longevity scenario of the pension scheme membership occurs; and(c) calculating, using computing apparatus and using the calculations ofthe expected payment schedule and a present value of the financialinstrument in the case of a stressed longevity scenario, an estimate ofthe longevity capital reserve required to ensure that the future cashflow obligations of the financial instrument would be covered in theevent that the stressed longevity scenario were to occur. The presentinvention also provides a computing apparatus configured to be operableto implement this method.

The method of invention is intended to be used in relation to afinancial instrument under which it is agreed that payments are to bemade that match or mirror the financial obligations of a pension scheme(for example, a defined benefit pension scheme related to a corporateentity) to at least a portion of its members. Payments under thefinancial instrument may be calculated in accordance with the methoddescribed herein. Such a ‘longevity’ financial instrument, tied to apension scheme, securitizes the longevity (and other) risks associatedwith the pension scheme by transferring them onto the capital markets.Investment in the financial instrument thereby achieves immunization ofthe longevity (and other) risks associated with that pension scheme forinvestors in the financial instrument. By investing in such a financialinstrument, a pension scheme can defease its longevity (and other risks)and prevent a deficit from growing. The longevity financial instrumentmay be any suitable financial instrument described herein by whichpayments are made in amounts related to the financial obligations of apension scheme to its members on any agreed basis (including an agreedlongevity scenario).

In accordance with the method of the invention the capital requirementto ensure that a longevity-based risk source in the longevity financialinstrument is covered up to a desired level can be estimated. Thisestimation is generally achieved in the invention by performing stresstests to project how the obligations under the financial instrumentwould change from that which would be payable if an expected longevityscenario were to be experienced by the pension scheme membership to thatwhich would be payable if a stressed longevity scenario, in which alongevity shock of the desired level occurs, were to be experienced bythe pension scheme membership.

By expected longevity scenario it is meant, generally, the expectedlongevity of each pension scheme member whose payments under the pensionscheme are taken into account when determining the financial obligationsof the financial instrument. The expected longevity for each member willtypically be embodied by a mortality table for that member, theindividual mortality tables comprise the expected longevity ‘scenario’.However, any method by which the mortality of the pension scheme memberscan be projected can provide a longevity ‘scenario’.

The expected longevity scenario is that under which the expected paymentschedule of the financial instrument is calculated, typically by themethod described herein, and may be on any agreed or stated longevitybasis. However, the expected longevity scenario is typically thatproduced by a longevity projection model of the longevity of the pensionscheme membership. The longevity projection model may be produced by themethod described herein, in which the model is based on standardmortality tables for a suitable reference population, being adjusted, ata group level, for underlying mortality trends in the referencepopulation and, at an individual or ‘granular’ level, mortality leveladjustments take into account socio-economic and other factors affectingthe longevity of the individual pension scheme members. By modelling thelongevity of the pension scheme membership in this way to produce anexpected longevity scenario, the longevity risk sources in the financialinstrument are reduced compared to a non-modelled longevity scenario, ornon-trend and non-level adjusted longevity model.

Whatever scenario is chosen as the expected longevity scenario, themethod of the invention can be used to estimate a capital reserverequirement to ensure that the obligations under the financialinstrument are covered in the event that, instead of the expectedlongevity scenario occurring, a different, more penal (i.e. expensive),‘stressed’ longevity scenario actually occurs. The stressed longscenario may be chosen on any basis. For example, a simple percentagemortality rate difference from the expected longevity scenario couldprovide a stressed longevity scenario. However, to provide a moremeaningful quantification of the longevity risk exposure of thefinancial instrument, a more methodical approach is typically taken toanalysing the longevity risk and producing a resultant stressedlongevity scenario. The risk capital requirement from various longevityrisk sources can, for example, be calculated separately.

By calculating a present value of the financial instrument in thestressed longevity scenario in accordance with the invention, theassociated capital requirement can be calculated, typically bycomparison with a present value of the financial instrument in theexpected longevity scenario. Note that a stressed payment schedule doesnot necessarily need to be calculated to assess the effect of thelongevity related shock in the stressed longevity scenario on thepresent value of the financial instrument. Instead, in accordance theinvention, the effect of the longevity related shock may be evaluated byobserving its effect directly on a present value of the financialinstrument.

By the issuer or underwriter holding the amount of risk capitalestimated by the method of the invention, the financial instrument canbe said to be supported up to that level of longevity risk (i.e. thedifference between the expected and stressed longevity scenarios).

Using the Inventive Method to Size ‘Trend’ Risk

The capital requirement to cover the different longevity risk sourcesmay be assessed by performing the method of the invention in relation tothe different longevity risk sources separately.

To estimate the capital reserve required to cover a mortality ‘trend’shock, in accordance with the above aspect of the invention, thelongevity-related shock is taken to be a mortality trend shock to theexpected longevity scenario, such that the longevity capital reserverequirement estimated by the model is that required to cover thelongevity risk exposure of the financial instrument in the case of afuture mortality trend shock occurring.

Here, mortality ‘trend’ shock and longevity ‘trend’ shock may be usedinterchangeably to indicate a future shock in which the mortality trendunderlying the expected longevity scenario (i.e. thoseimprovements/changes in mortality that are expected to occurcollectively across the pension scheme membership at a ‘group’ level andalso more broadly in the general reference population) turn out to bedifferent to the underlying mortality trends that are actuallyexperienced at a group level (and also more broadly in the generalreference population). Such factors that affect the underlying mortalitytrends of the pension scheme membership and more broadly in the generalpopulation are typically: public health standards; general improvementsin the level of medical care provision; public health disasters anddisease epidemics; the development of cures or treatments for particularterminal conditions, etc.

In accordance with this method, the size of the ‘trend’ risk exposure inthe financial instrument can be quantified and analysed.

To achieve this trend risk analysis, step (b) of the method of theinvention may comprise: calculating, using computing apparatus, astressed payment schedule of the financial instrument by calculatingwhat the cash flow obligations of the pension scheme would be in thecase of a stressed longevity scenario for the pension scheme membershipin which a mortality trend shock to the expected longevity scenario ofthe pension scheme membership occurs; and calculating, using computingapparatus, a present value of the financial instrument in the case ofthe stressed payment schedule occurring.

To deterministically estimate the size the capital reserve required tocover a longevity trend shock of a given magnitude (i.e. of any assumed,possibly non-modelled, group ‘trend’ shock), in step (b) of the method,the stressed longevity scenario may be selected on the basis of alongevity scenario incorporating a given mortality trend shock, and, instep (c), the estimate of the longevity capital reserve requirement maybe calculated as the difference between a present value of the stressedpayment schedule and a present value of the expected payment schedule.

Typically, the longevity of the pension scheme membership will, however,be modeled, and, to achieve this, step (a) may comprise: modelling,using computing apparatus, the longevity of the pension schememembership to produce a longevity projection model for the pensionscheme which provides longevity projections for the pension schememembership; and selecting the expected longevity scenario as the bestestimate longevity projection of the pension scheme membership providedby the longevity projection model. Further, the longevity projectionmodel may take into account mortality tables for a suitable referencepopulation adjusted for: at a group level, underlying mortality trendsmodelled in the suitable reference population; and/or at an individuallevel, mortality level adjustments corresponding to each pension schememember's socio-economic characteristics.

To accurately apply longevity trend shocks in the method over a giventime horizon and determine the effect of the shock of the paymentschedule of the financial instrument, in step (b), the stressedlongevity scenario may be constructed to comprise: over the duration ofa given time horizon, a shocked longevity projection of the pensionscheme membership incorporating a mortality trend shock; and after thegiven time horizon, a best estimate longevity projection of the pensionscheme membership output by the longevity projection model, the modelhaving been updated to take into account the experience of the mortalitytrend shock over the duration of the time horizon.

To provide a meaningful and useful deterministic assessment of the sizeof the longevity trend risk, a means of probabilistic modelling of trendrisk is included in the model, such that the longevity projection modelmay further provide confidence intervals around the best estimatelongevity projection by which the probability of a longevity projection,incorporating a mortality trend shock given by the confidence intervals,occurring can be estimated, and wherein, in step (b), the shockedlongevity projection of the stressed longevity scenario may be selectedon the basis of a confidence interval longevity projection of thelongevity projection model, the shocked longevity projection beingprojected to occur with a probability estimated according to thelongevity projection model. Thus by probabilistic modelling of thelongevity of the pension scheme membership, longevity projections inwhich a longevity trend shock occurs can be modelled and said to belikely to occur with a certain probability predicted by the model. Thesetrend-shocked longevity projections produced by the model can be used inaccordance with the method to build up a specific stressed longevityscenario and deterministically estimate the longevity capitalrequirement to cover the a trend shock projected to occur with a certainprobability.

In view of this probabilistic modelling, step (b) may include selectingthe shocked longevity projection as being that longevity projected tooccur by the longevity projection model with a given probability. Bythis method, an estimate of the longevity capital reserve required tocover a longevity trend shock that is modelled to occur with a givenprobability (e.g. a future trend shock event with a 1% likelihood ofoccurring) can be provided. Step (c) may comprise calculating theestimate of the longevity capital reserve requirement as the differencebetween a present value of the stressed payment schedule and a presentvalue of the expected payment schedule.

The method of the invention may be used to estimate the capitalrequirement to achieve a given rating from a ratings agency, for exampleStandard & Poors or Fitch. To achieve this, the stressed longevityscenario may calibrated to a probability given by a rating agency'sdefault probability rate table, whereby the shocked longevity projectionof the stressed longevity scenario is selected to be that projected bythe longevity projection model to occur with a probability of no morethan the default probability of a corporate bond having a given ratingaccording to the default probability rate table. By choosing the ‘given’probability in accordance with a rating agency's default probabilityrate table in this way, the capital requirement (at least in respect ofthe longevity trend risk) to achieve a given rating can be estimated.

Longevity trend shocks can be experienced over different time horizons,depending on when they are applied in the construction of the stressedlongevity scenario. To ensure that the estimated capital requirement issufficient to cover the longevity risk up to a given level for all timehorizons, the method may further comprise: repeating step (b) to performa plurality of separate stress tests each providing a stressed paymentschedule on the basis of a different stressed longevity scenario, eachdifferent stressed longevity scenario comprising a shocked longevityprojection incorporating a mortality trend shock over one of a pluralityof different time horizons, the shocked longevity projection of eachstressed longevity scenario being calibrated to the probability given bythe rating agency's default probability rate table for the same ratingcorporate bond and for a tenor matching the relevant time horizon forthe relevant mortality trend shock; and step (c) may then comprise:calculating a present value of each of the plurality of stressed paymentschedules; identifying the worst (i.e. most penal/expensive) stressedpayment schedule having the highest present value; and calculating theestimate of the longevity capital reserve requirement as the differencebetween the present value of the worst stressed payment schedule and apresent value of the expected payment schedule. By performing stresstests applying longevity shocks over all different time horizons, thecapital required to ensure that the financial instrument is supported tothe desired trend shock level/credit rating (at least in respect oftrend shocks) can be calculated.

In the above deterministic approaches to sizing longevity trend risk, alongevity trend shock of an appropriate size (which may be projected bya model to occur with a certain probability) is applied to the expectedlongevity scenario for the pension scheme membership over a given timehorizon, and the effect on the payment schedule and a present valuethereof is observed.

Alternatively, a stochastic approach to sizing longevity trend risk maybe taken, in which a series of random longevity trend shocks are appliedto the longevity projections for the pension scheme membership over asequence of time horizons is repeatedly simulated, and for eachsimulation the effect on the payment schedule and a present valuethereof is observed to provide a probability distribution of presentvalues for the financial instrument. From this simulated distribution,the longevity trend risk capital requirement can be estimated.

To achieve this stochastic method in the invention, step (b) may includerepeatedly stochastically simulating a stressed payment schedule for thefinancial instrument in which simulation a sequence of random mortalitytrend shocks are applied at given time horizons, each mortality trendshock being applied by randomly selecting, at a given time horizon, theshocked longevity projection in accordance with a probabilitydistribution for the pension scheme membership longevity projectionmodelled by the confidence intervals of the longevity projectionprovided by the longevity projection model, the repeated simulationsproviding, for each given time horizon, a plurality of simulatedstressed payment schedule outcomes each being associated with a randomlystressed longevity scenario.

Further, step (c) may comprise calculating, for each given time horizon,a present value of each of the plurality of simulated stressed paymentschedule outcomes, and generating, for each given time horizon, aprobability distribution of present values of stressed paymentschedules. By this method, a probability distribution of possiblepresent values for the financial instrument as a result of theuncertainty associated with the longevity trend risk exposure of thefinancial instrument, can be calculated.

In accordance with this stochastic method, it is possible to estimatethe capital reserve required to ensure that longevity trend risk iscovered in a range of possible outcomes up to that projected to occurwith a given probability. To achieve this, step (c) may comprise:determining, for a given time horizon, a present value of a stressedpayment schedule which, from the probability distribution of presentvalues of stressed payment schedules for that time horizon, is simulatedto occur with a given probability; and calculating the estimate of thelongevity capital reserve requirement as the difference between thepresent value of the stressed payment schedule that is simulated tooccur with the given probability and a present value of the expectedpayment schedule.

As with the deterministic method above, the stochastic method of theinvention may be used to estimate the capital requirement to achieve agiven rating from a ratings agency, for example Standard & Poors orFitch. To achieve this, in step (c), the calculation of the estimate ofthe longevity capital reserve requirement may be calibrated to aprobability given by a rating agency's default probability rate table,whereby step (c) may comprise: determining, for a given time horizon, apresent value of a stressed payment schedule which, from the probabilitydistribution of present values of stressed payment schedules for thattime horizon, is simulated to occur with a probability of no more thanthe default probability of a corporate bond having a given ratingaccording to the default probability rate table for a tenor matching therelevant time horizon; and wherein step (c) may further comprise:calculating the estimate of the longevity capital reserve requirement asthe difference between the present value of the stressed paymentschedules that is simulated to occur with the given probability and apresent value of the expected payment schedule.

Longevity trend shocks can be experienced over different time horizons,depending on when they are applied in the construction of the stressedlongevity situation. To ensure that the estimated capital requirement issufficient to cover the longevity risk up to a given level for all timehorizons, under the stochastic approach, in step (c), the calculation ofthe estimate of the longevity capital reserve requirement may becalibrated to a probability given by a rating agency's defaultprobability rate table, and step (c) may comprise: determining, for eachtime horizon, a present value of a stressed payment schedule which, fromthe respective probability distribution of present values of stressedpayment schedules for that time horizon, is simulated to occur with aprobability of no more than the default probability of a corporate bondhaving a given rating according to the default probability rate tablefor a tenor matching the relevant time horizon; selecting, from acrossall time horizons, the worst present value of the stressed paymentschedules that is simulated to occur with the given probability; andcalculating the longevity capital reserve requirement as the differencebetween the worst present value of the stressed payment schedules and apresent value of the expected payment schedule.

As the stochastic approach provides a range of possible present valuesof the financial instrument under simulated longevity trend shocks (as aprobability distribution), instead of analysing the capital reserverequirement to cover longevity trend risk in relation to a longevitytrend shock projected to occur with a given probability, it is insteadpossible to perform the analysis of capital reserve requirement toensure a given expected loss on the financial instrument due tolongevity trend risk. To achieve this, step (c) may comprisescalculating, on the basis of the probability distribution of presentvalues of stressed payment schedules for a given time horizon, theamount of longevity capital reserve required to ensure that the expectedloss on the financial instrument for that time horizon is no greaterthan a given expected loss.

As a result, the stochastic method of the invention may be used toestimate the capital requirement to achieve a given rating from aratings agency, such as Moody's, which assesses the rating of afinancial instrument in relation to the expected losses on thatfinancial instrument. To achieve this, in the invention, the longevitycapital reserve requirement calculation is calibrated to a ratingagency's idealized loss rate table, whereby the given expected loss isan expected loss given by the idealized loss rate table.

In accordance with this expected loss-based method, the calculation ofthe longevity capital reserve requirement may include calculating, for agiven amount of capital held, the expected loss on the financialinstrument, where the calculation of the expected loss may comprise:fitting a model distribution to the probability distribution of presentvalues of stressed payment schedules for the given time horizon;calculating, from the fitted distribution, a probability of a shortfalloccurring given the amount of capital held; and calculating, from thefitted distribution, an estimate of the average shortfall given theamount of the capital held and calculating the loss given shortfall asthe ratio of the average shortfall to the present value of the expectedpayment schedule; where the calculation of the expected loss may includemultiplying the probability of a shortfall by the loss given shortfall.

The calculation of the longevity capital reserve requirement may includeiteratively performing calculations of the expected loss for differentgiven amounts of capital held to determine the amount of capitalrequired to ensure that the expected loss is not less than the givenexpected loss. The calculation of the probability of a shortfalloccurring may include calculating, from the fitted distribution, theprobability that the present value of the stressed payment schedule willexceed the present value of the expected payment schedule by the givenamount of capital held; and the calculation of the average shortfallgiven the amount of capital held may include taking an average of arandom sample of present values in the tail of the fitted distributionin the region of present values greater than the sum of the presentvalue of the expected payment schedule by and the given amount ofcapital held.

In accordance with the above methods, the size of difference tranches oflongevity risk capital associated with the financial instrument can beestimated. For example, in accordance with the Standard & Poor's defaultprobability rating table, the capital requirement of the tranches oflongevity trend risk rated AAA, AA, A and BBB may be estimated using themethod of the invention.

Using the Inventive Method to Size ‘Process’ Risk

The inventors have identified that another longevity-related risk sourcefor the longevity financial instruments is a ‘process’ risk that, evenif the expected longevity scenario did reflect the ultimate longevityunderlying the pension scheme membership, there would, due to the finitesample size of the pension scheme membership, be a resultant samplingerror that leads to an inherent distribution of possible outcomes. Thiserror introduces a process risk that the specific pension schememembership will, by chance, and due to the small sample size, live, onaverage, longer than expected in the expected longevity scenario. Themagnitude of this risk source is related to this size of the pensionscheme membership (i.e. the ‘sample’), the larger the pension scheme thelower the sampling error and the less significant the process risk.Nevertheless, process risk can be a significant risk source and so tosize the risk capital requirement to cover the process risk associatedwith a pension scheme up to a given level, the inventive method of theabove aspect may be used. To achieve this, the longevity-related shock(of step (b) of the method) is taken to be a shock resulting from asample error resulting from the size of the pension scheme membership,such that the longevity capital reserve requirement estimated by themethod is that required to cover the longevity risk exposure of thefinancial instrument in the case of the longevity of the pension schememembership being different from the expected longevity scenario due to asample error occurring.

To achieve this capital requirement estimation, step (b) may comprise:characterising, by performing a bootstrapping analysis on a referencepopulation using data processing apparatus, a sampling errordistribution for the longevity projections according to an expectedlongevity scenario for the reference population, the sampling errordistribution being associated with a size of the population of thepension scheme; and applying, using data processing apparatus, saidsampling error distribution to a present value of the expected cashflows to produce a distribution of present values of the financialinstrument occurring in the case of a sample error; and step (c) maycomprise determining the amount of longevity risk capital required toensure that: the payment schedule on the financial instrument would becovered in the case of a sample error in the longevity projections whichis projected to occur by the bootstrapping analysis with a givenprobability; or the expected loss that would result from a sample errorin the longevity projections is lower than a given expected loss. Thusin accordance with the method, the capital reserve required to cover thelongevity ‘process’ risk can be calculated for a sample error of a givenprobability, or for a given resulting expected loss.

The given probability may be the default probability of a bond having anequivalent rating according to a rating agency's default probabilityrate table. The given expected loss may be the expected loss of a bondhaving an equivalent rating according to a rating agency's idealisedloss rate table. Thus the calculation of the reserve required to coverthe longevity ‘process’ risk can be calibrated to a rating agencyrequirement and thus be used to assess the capital requirement toachieve a rating from a rating agency (at least in respect of the‘process’ risk).

The bootstrapping analysis may comprise: calculating, for N randomsamples of members of the reference population of the same size as thepopulation of the pension scheme, the projected mortality rate accordingto the expected longevity scenario for the reference population for thatrandom sample for a period of time; comparing, for each sample, theprojected mortality rate with the actual mortality rate for that sampleof the reference population and for that period of time to determineerrors in the mortality rate projections; and characterising thedistribution of the errors in the longevity projections of the longevityprojection model.

Using the Inventive Method to Size ‘Level’ Risk

The inventors have identified that, despite the granular mortality leveladjustments made to the mortality tables underlying the projection ofthe pension scheme liabilities in their methodology, there remains inthe longevity financial instruments an inherent exposure to mortalitylevel risk—i.e. the risk that the mortality level adjustments made arenot correct. The inventors have therefore devised a method ofquantifying the mortality level risk exposure of the financialinstruments. This method can be used even where no mortality level riskadjustments are made in the expected longevity scenario for the pensionscheme membership.

To achieve this, the longevity-related shock (of step (b) of the method)is taken to be a mortality level shock to the expected longevityscenario of the pension scheme membership, such that the longevitycapital reserve requirement estimated by the model is that required tocover the mortality level risk exposure in the financial instrument inthe case of a future mortality level shock occurring.

More specifically, to achieve this, step (b) may comprise: determiningthe actual number of deaths already occurred in an available historicaldata set for the pension scheme; modelling, for the historical data setand using data processing apparatus, a probability density function forthe number of deaths in the historical data set based on the mortalitytables for a suitable reference population; determining, using dataprocessing apparatus, a shocked mortality level adjustment which, whenapplied to the mortality tables in the said model, provides aprobability density function for which the probability of the actualnumber of deaths occurring is equal to a given probability, the stressedlongevity scenario being constructed by applying the shocked mortalitylevel adjustment to the mortality tables; and calculating, using dataprocessing apparatus, a stressed payment schedule of the financialinstrument by calculating what the cash flow obligations of the pensionscheme would be in the case of the stressed longevity scenario occurringand calculating the present value of the financial instrument in thestressed longevity scenario on the basis of the stressed paymentschedule. Thus in accordance with this method, the actual historicaldata for the pension scheme membership can be used to determine ashocked mortality ‘level’ adjustment which is projected to occur with agiven probability. From this a present value of the financial instrumentin this stressed longevity scenario including the shocked mortality‘level’ adjustment can be calculated, which then allows the capitalreserve requirement to cover the shocked mortality level adjustment tobe calculated.

The shocked mortality level adjustment may be calibrated to aprobability given by a rating agency's default probability rate tablesuch that the actual number of deaths is projected to occur with aprobability of no more than the default probability of a corporate bondhaving a given rating according to the default probability rate table.In accordance with this method the capital reserve requirement to coverthe shocked mortality level adjustment and achieve a rating from aratings agency can be estimated.

The model of the probability density function of scheme member deathsmay assume that deaths follow a binomial distribution for a given age.The model of the probability density function of scheme member deathsmay be based on a Poisson approximation or on a Monte Carlo simulation.

Alternatively, a Value at Risk approach may be taken to assessing thecapital reserve requirement to cover a shocked mortality leveladjustment. To achieve this, step (b) may comprise: determining theactual number of deaths already occurred in the available historicaldata for the pension scheme; modelling, for the historical data set andusing data processing apparatus, a probability density function for thenumber of deaths in the data set based on the mortality tables for asuitable reference population; and calculating, using data processingapparatus, the value at risk in a present value of the payment scheduleamounts of the financial instrument due to a shocked mortality leveladjustment for which, when applied to the expected longevity scenariofor the pension scheme membership, the actual level of deaths in thehistorical data is modelled to occur with a given probability, the valueat risk being used in step (c) to determine the longevity capitalreserve requirement.

Viewed from another aspect, the present invention provides acomputer-implemented method of estimating a capital reserve requirementto cover the longevity risk exposure of a financial instrument in thecase of a future mortality trend shock, the financial instrumentundertaking to pay to an investor sums according to a payment scheduleof amounts arranged to match with the future cash flow obligations of apension scheme to at least a portion of its members; the methodcomprising: (a) calculating, using computing apparatus, an expectedpayment schedule of the financial instrument by calculating what thecash flow obligations of the pension scheme to its relevant memberswould be in the case of an expected longevity scenario for the pensionscheme membership occurring; (b) calculating, using computing apparatus,a stressed payment schedule of the financial instrument by calculatingwhat the cash flow obligations of the pension scheme would be in thecase of a stressed longevity scenario for the pension scheme membershipin which a mortality trend shock to the expected longevity scenario ofthe pension scheme membership occurs; and (c) calculating, usingcomputing apparatus and using the calculations of the expected paymentschedule and the stressed payment schedule, an estimate of the longevitycapital reserve required to ensure that the future cash flow obligationsof the financial instrument would be covered in the event that thestressed longevity scenario were to occur. The present invention alsoprovides a computing apparatus configured to be operable to implementthis method.

Viewed from still another aspect, the present invention provides acomputer-implemented method of estimating a capital reserve requirementto cover the longevity risk exposure of a financial instrument in thecase of a future mortality level shock, the financial instrumentundertaking to pay to an investor sums according to a payment scheduleof amounts arranged to match with the future cash flow obligations of apension scheme to at least a portion of its members; the methodcomprising: (a) calculating, using computing apparatus, an expectedpayment schedule of the financial instrument by calculating what thecash flow obligations of the pension scheme to its relevant memberswould be in the case of an expected longevity scenario for the pensionscheme membership occurring; (b) calculating, using computing apparatus,a present value of the financial instrument in the case of a stressedlongevity scenario for the pension scheme membership in which amortality level shock to the expected longevity scenario of the pensionscheme membership occurs; and (c) calculating, using computing apparatusand using the calculations of the expected payment schedule and apresent value of the financial instrument in the case of a stressedlongevity scenario, an estimate of the longevity capital reserverequired to ensure that the future cash flow obligations of thefinancial instrument would be covered in the event that the stressedlongevity scenario were to occur. The present invention also provides acomputing apparatus configured to be operable to implement this method.

Viewed from still another aspect, the present invention provides acomputer-implemented method of estimating a capital reserve requirementto cover the longevity risk exposure of a financial instrument in thecase of a sample error, the financial instrument undertaking to pay toan investor sums according to a payment schedule of amounts arranged tomatch with the future cash flow obligations of a pension scheme to atleast a portion of its members, the sample error resulting from the sizeof the pension scheme membership; the method comprising: (a)calculating, using computing apparatus, an expected payment schedule ofthe financial instrument by calculating what the cash flow obligationsof the pension scheme to its relevant members would be in the case of anexpected longevity scenario for the pension scheme membership occurring;(b) calculating, using computing apparatus, a present value of thefinancial instrument in the case of a stressed longevity scenario forthe pension scheme membership in which a sample error resulting from thesize of the pension scheme membership occurs; and (c) calculating, usingcomputing apparatus and using the calculations of the expected paymentschedule and a present value of the financial instrument in the case ofa stressed longevity scenario, an estimate of the longevity capitalreserve required to ensure that the future cash flow obligations of thefinancial instrument would be covered in the event that the stressedlongevity scenario were to occur. The present invention also provides acomputing apparatus configured to be operable to implement this method.

Assessing the Total Capital Requirement Taking into AccountDiversification Benefit

The inventors have also identified that the various risk sourcesinherent in the longevity financial instruments of their methodologythat each contribute to a risk capital requirement to support thefinancial instrument may combine to produce a diversification benefitthat effectively reduces the total risk capital requirement. It would beadvantageous to gain from this benefit as the financing required tosupport the longevity financial instrument would then be reduced.

Therefore, viewed from another aspect, the present invention provides acomputer-implemented method of calculating, for a financial instrumentthat provides to investors at least a partial hedge against longevityrisk exposure in at least a portion of a specific pension scheme, thereduction in the total risk capital required to support the financialinstrument due to the diversification benefit of the longevity risksources and market risk sources to which the financial instrument isexposed, the method comprising: constructing, using data processingapparatus, a correlation matrix of assumed correlation coefficients, ρ,between each pair of risk sources; and calculating, using dataprocessing apparatus, a diversified risk capital requirement, C_(total),on the basis of using the following equation:

C _(total)=√{square root over (ΣC _(i) ²Σρ_(ij) C _(i) C _(j))}

wherein C_(i) is the capital requirement for risk source i and ρ_(ij) isthe assumed correlation coefficient between risk sources i and j;wherein the reduction in the total risk capital requirement due todiversification benefit is the difference between C_(total) and the sumof C_(i) across all risk sources i. The present invention also providesa computing apparatus configured to be operable to implement thismethod.

In accordance with this aspect of the invention, the diversificationbenefit of the combined effects of the various risk sources, such aslongevity risk (including longevity trend risk and mortality level risk)and the various other economic risks, can be quantified. Using thismethod, the total capital requirement can then be reduced by the amountgiven by the diversification benefit of the risks without the creditrating of the financial instrument being impaired (i.e. the probabilityof default or the expected loss increasing) or the total value at riskincreasing.

Preferably the model of the probability density function of schememember deaths is based on a Monte Carlo simulation.

As an alternative to the above variance-covariance method of calculatingthe diversification benefit of the various longevity and market risksources, the inventors have also developed a Monte Carlo simulationmethod, which in some cases is used in preference to thevariance-covariance method to calculate the diversification benefit.

Therefore, viewed from another aspect, the present invention provides acomputer-implemented method of calculating, for a financial instrumentthat provides to investors at least a partial hedge against longevityrisk exposure in at least a portion of a specific pension scheme, thetotal risk capital required to support the financial instrument takinginto account the diversification benefit of the longevity risk sourcesand market risk sources to which the financial instrument is exposed,the method comprising: repeatedly simulating, using a Monte Carlo methodand a data processing apparatus, the projected losses on the financialinstrument due to each risk source to produce a number of outcomes;calculating for each outcome, using a data processing apparatus, acombined projected loss as the sum of the projected losses due to eachrisk source; and calculating, using a data processing apparatus, thetotal risk capital requirement on the basis of the distribution ofoutcomes of the combined projected losses. The present invention alsoprovides a computing apparatus configured to be operable to implementthis method.

In accordance with this aspect of the invention, the total risk capitalrequirement, which incorporates the diversification benefit of thecombined effects of the various risk sources, such as longevity risk(including longevity trend risk and mortality level risk) and thevarious other economic risks, can be calculated.

Preferably, the total risk capital requirement is calculated as thatwhich would result in a combined projected loss that is simulated tooccur with a probability equal to the probability of default of a bondhaving an desired credit rating according to a credit rating agency'sdefault probability rate table for an appropriate time horizon matchedto the duration of the pension scheme's liabilities.

Preferably, the total risk capital requirement is calculated as thatwhich would ensure the combined projected loss is less than the expectedloss of a bond having a desired credit rating according to a creditrating agency's idealised loss rate table for an appropriate timehorizon matched to the duration of the pension scheme's liabilities.

In accordance with these preferred arrangements, the total risk capitalrequirement to achieve a desired credit rating for the financialinstrument can be estimated, including the diversification benefit.

Preferably, there is assumed in the Monte Carlo simulations andsubsequent calculations to be no correlation between the risk sources.

Viewed from yet another aspect, the present invention provides a methodof achieving a rating for the longevity risk exposure of a financialinstrument provided to an investor, the instrument undertaking to pay,at regular points in time over a specified duration, sums according to aschedule of payment amounts associated with the financial instrument,said scheduled payment amounts being arranged to match with the expectedcash flow obligations of a pension scheme to its members, said expectedcash flow obligations at a point in time being calculated at leasttaking into account the projected likelihood that each pension schememember will survive until that point in time, the method comprising:calculating, by performing stress tests on the expected cash flows, theamount of risk capital required to be held to achieve the rating and toensure that the expected loss that would result from a mortality shockis lower than the expected loss of a bond having an equivalent creditrating according to a credit rating agency's idealised loss rate table;and holding said amount of risk capital.

Viewed from yet another aspect, the present invention provides a methodcomprising: providing to an entity a financial instrument whichundertakes to pay to the entity, at regular points in time within aspecified duration, sums according to a schedule of payment amountsassociated with the financial instrument, said scheduled payment amountsbeing arranged to match with expected cash flow obligations of a pensionscheme to members of the pension scheme; and at a re-set point in time,resetting the schedule of payment amounts such that the entity willreceive an adjusted payment amount at a scheduled time calculated to bean aggregate of nominal cash flows to be paid to the members of thepension scheme adjusted to take into account actual cumulative mortalityexperience within the pension scheme prior to the re-set point in time;and calculating, by performing longevity stress tests on the expectedcash flows using data processing apparatus, an amount of longevity riskcapital to be held, and holding said amount of risk capital.

The method may further comprise holding the required amount of capital.The financial instrument may carry a rating from a rating agency. Thefinancial instrument may carry a rating from at least one of Standard &Poor's, Moody's and Fitch rating agencies.

The longevity risk capital may be raised by issuing subordinatedtranches of debt and equity capital in the form of capital notes andequity notes. The subordinated tranches of capital notes and equitynotes may further have an exposure to asset risk.

The financial instrument may carry a rating from a rating agency and asubordinated tranche of capital may be sized to have a capitalisationthat corresponds to a junior rating from the rating agency and bepositioned accordingly in a sequential payment structure of a paymentwaterfall.

For a scheduled payment point in time said expected cash flowobligations of the pension scheme to its members may be calculatedtaking into account at least the projected likelihood that each pensionscheme member will survive until that point in time, and the financialinstrument may carry a rating from a rating agency, the rating havingbeen achieved by performing stress tests on the expected cash flows soas to calculate an amount of risk capital required to be held to achievethe rating and to ensure that the payment amount obligations on thefinancial instrument can be met in the case of a worst case longevityshock which is projected to occur with a probability of no more than thedefault probability of a bond having an equivalent rating according to arating agency's default probability rate table. The rating agency may beStandard & Poors or Fitch.

The projected likelihood that each pension scheme member will survivemay be calculated by modelling changes in the probability of survival ofa suitable reference population by using a statistical longevityprojection model to extrapolate, into the future, trends in the actualmortality experience of that reference population and adjusting themortality table associated with the reference population to incorporatethese trends, and the stress tests on the expected cash flows may beperformed by using the statistical longevity projection model tosimulate the Net Present Value of the pension scheme's expected cashflow obligations for varying future longevity outcomes. The method mayincluding determining, for a plurality of future points in time and forall varying future longevity outcomes, a binding time horizon having alargest Net Present Value of the pension scheme's expected cash flowobligations that is projected to occur with a probability of no morethan the default probability of a bond having an equivalent ratingaccording to a rating agency's default probability rate table; and therisk capital requirement may be the difference between said largest NetPresent Value of the pension scheme's expected cash flow obligations anda best estimate of said Net Present Value at said binding time horizon.

Alternatively, for a scheduled payment point in time said expected cashflow obligations of the pension scheme to its members may be calculatedtaking into account at least the projected likelihood that each pensionscheme member will survive until that point in time, and the financialinstrument may carry a rating from a rating agency, the rating havingbeen achieved by performing stress tests on the expected cash flows tocalculate the amount of risk capital required to be held to achieve therating and to ensure that the expected loss that would result from amortality shock is lower than the expected loss of a bond having anequivalent credit rating according to a credit rating agency's idealisedloss rate table. The credit rating agency may be Moody's.

The projected likelihood that each pension scheme member will survivemay be calculated by modelling changes in the probability of survival ofa suitable reference population by using a statistical longevityprojection model to extrapolate, into the future, trends in the actualmortality experience of that reference population and adjusting themortality table associated with the reference population to incorporatethese trends, and the stress tests on the expected cash flows may beperformed by using the statistical longevity projection model tosimulate a Net Present Value of the pension scheme's expected cash flowobligations for varying future longevity outcomes. The method mayinclude determining, for a plurality of future time points and for allvarying future longevity outcomes, a binding time horizon having thelargest Net Present Value of the pension scheme's expected cash flowobligations that results in an expected loss of a bond having anequivalent rating according to a rating agency's idealised loss ratetable; the risk capital requirement may be the difference between saidlargest Net Present Value of the pension scheme's expected cash flowobligations and a best estimate of said Net Present Value at saidbinding time horizon.

The calculation of the amount of risk capital required to be held toachieve the rating specified by the rating agency may further comprises:characterising, by performing a bootstrapping analysis on a referencepopulation, an error distribution for the mortality projections producedby a statistical mortality projection model, the error distributionbeing associated with a size of the population of the pension scheme;and determining, by applying said error distribution to the Net PresentValue of the expected cash flows, the amount of risk capital to be heldto ensure that: the payment amounts on the financial instrument can bemet in the case of a sample error in the mortality projections which isprojected to occur with a probability of no more than the defaultprobability of a bond having an equivalent rating according to therating agency's default probability rate table; or the expected lossthat would result from a sample error in the mortality projections islower than the expected loss of a bond having an equivalent ratingaccording to the rating agency's idealised loss rate table.

The bootstrapping analysis may comprise: calculating, for N randomsamples of members of the reference population of the same size as thepopulation of the pension scheme, the mortality rate projected by thestatistical mortality projection model for that random sample for aperiod of time; comparing each of said mortality rate projections withthe actual mortality rate for that sample of the reference populationand for that period of time to determine errors in the mortalityprojections; and characterising the distribution of the errors in themortality projections.

Viewed from yet another aspect, the present invention provides acomputer-implemented method of calculating, in relation to a financialinstrument that pays to an investor a cash flow according to a paymentschedule arranged to match with at least part of the cash flowobligations of a pension scheme to at least a portion of its memberssuch that the financial instrument provides to the investor at least apartial hedge against longevity risk exposure in said pension scheme, anamount of risk capital required to support longevity trend risk exposurein the financial instrument and achieve a specific rating for thefinancial instrument, the method comprising: calculating, using dataprocessing apparatus, the expected cash flow of the financial instrumentat a time horizon at least taking into account the projected likelihoodthat each pension scheme member will survive until that time horizon,calculating, using data processing apparatus to perform stress tests onthe expected cash flows of the financial instrument, an amount of riskcapital required to be held to achieve the rating and to ensure that thepayment amount obligations on the financial instrument can be met in thecase of a worst case longevity shock which is projected to occur with aprobability of no more than the default probability of a bond having anequivalent rating according to a rating agency's default probabilityrate table; and holding at least said amount of risk capital.

Viewed from yet another aspect, the present invention provides a methodof quantifying the longevity risk exposure of a financial instrumentprovided to an investor, the instrument undertaking to pay, at points intime over a specified duration, sums according to a schedule of paymentamounts associated with the financial instrument, said scheduled paymentamounts being arranged to match with the expected cash flow obligationsof a pension scheme to its members, said expected cash flow obligationsat a point in time being calculated at least taking into account theprojected likelihood that each pension scheme member will survive untilthat point in time; the method comprising: calculating a present valueof the financial instrument from the expected cash flows making up thepayment schedule; and calculating, by performing stress tests on theexpected cash flows, the change in the present value of the financialinstrument that occurs due to a longevity shock that is projected tooccur with the specified probability, the change in present valuerepresenting the risk exposure of the financial instrument to thelongevity shock having that probability.

Viewed from yet another aspect, the present invention provides a methodof quantifying the longevity risk exposure of an asset or a liability toa longevity shock that is projected to occur with a specifiedprobability, the asset or liability having cash flows of sums ofaccounts receivable and accounts payable at regular points in time overa specified duration, said sums being at least a function of the actualmortality experience of a group of creditors or debtors, comprising:calculating, for each point in time, the expected cash flows at thatpoint in time at least taking into account the projected likelihood thateach creditor or debtor will survive until that point in time;calculating a present value of the asset or liability from the expectedcash flows; and calculating, by performing stress tests on the expectedcash flows, the change in the present value of the asset or liabilitythat occurs due to a longevity shock that is projected to occur with thespecified probability, the change in present value representing the riskexposure of the asset or liability to the longevity shock having thatprobability.

The asset or liability may be selected from the group comprising: adefined benefit pension scheme; a defined contribution pension scheme;one or more equity release mortgages; one or more reverse mortgages; anda financial instrument that is arranged to transfer the longevity riskexposure of any of these assets and liabilities to the capital markets.

BRIEF DESCRIPTION OF THE DRAWINGS

Certain preferred embodiments of aspects of the present invention willnow be described by way of example only and with reference to theaccompanying drawings, in which:

FIG. 1 is a schematic drawing detailing a financial instrument andderivative product and the issuing entities according to one embodimentof the present invention;

FIG. 2 is a schematic drawing showing the interaction between theparties involved in the securitization of a pension scheme according toone embodiment of the present invention;

FIG. 3 illustrates a data processing system for use in carrying outmethods in accordance with the invention;

FIG. 4 shows a flow chart of data transfer and feed-through for thevarious modules comprising the pensions securities trading and reportingsystem (risk management system) of the present invention;

FIG. 5 is a schematic drawing showing the hierarchy of tranches ofpensions defeasance products used in the method of securitizing apension scheme according to one embodiment of the present invention;

FIG. 6 is an illustration of the members comprising an exemplary pensionscheme to be defeased by a pensions defeasance product in accordancewith an embodiment of the invention;

FIG. 7 shows a projection of each exemplary pension scheme member'snominal cash flow;

FIG. 8 shows a prediction each exemplary pension scheme member'sexpected cash flow, taking account of their probability of death;

FIG. 9 shows the effect on each exemplary pension scheme member'spension value and nominal cash flow at year 10 due to a variation fromthe expected RPI value and a scheme member commuting a portion of hispension on retirement;

FIG. 10 shows the segmentation of each exemplary pension scheme member'scash flow at year 10;

FIG. 11 shows the allocation of the revised nominal cash flows at year10 to the segments of the exemplary pension scheme;

FIG. 12 shows the calculation of the average survival rate for eachsegment at year 10;

FIG. 13 shows the calculation of the year 10 indexed cash flow of theexemplary pension scheme and FIG. 14 shows the calculation of the year10 rate re-set;

FIG. 15 shows the calculation of what cash flows the trustees of theexemplary pension scheme actually need to pay their members and thesplitting the cash flows at year 10 into segments;

FIG. 16 shows the calculation of the basis risk exposure on the 10 yearrate re-set of the exemplary pension scheme;

FIG. 17 shows the annual percentage decline in mortality rate for malesaged 20-90 from the ONS data;

FIG. 18 shows the annual percentage decline in mortality rate for malesby age-group from the ONS data, illustrating the ‘cohort effect’;

FIG. 19 shows a comparison of P-Spline and CMI adjusted mortality rateprojections averaged for a representative ‘basket’ of males aged 55-90;

FIG. 20 shows a plot of the estimated default probabilities, which arederived from Standard & Poor's data for AAA, AA, A and BBB ratedcorporate bonds;

FIG. 21 shows an illustration of the calculation of the required capitalto cover different stresses based on the difference between the bestestimate liability value and the shocked liability value;

FIG. 22 shows the confidence intervals around the P-Spline Best Estimatefor the mortality rate projections for a 65-year-old male;

FIG. 23 shows an example of a shock scenario using a 5 year timehorizon;

FIG. 24 shows a calculation of shocked mortality rates for differenttime horizons for a single stochastic draw;

FIG. 25 shows the results of a stochastic simulation process;

FIG. 26 shows the calculation of Expected Loss;

FIG. 27 illustrates the main elements of the Longevity Capital Model(LCM);

FIG. 28 shows the distribution of the results from each of the 5,000simulations of the base case of the bootstrapping analysis forquantifying process risk;

FIG. 29 shows the effect on process risk of different pensionscheme/sample sizes for a sample size of 50,000 lives and 100,000 lives;

FIG. 30 illustrates the unknown nature of a probability distributionunderlying an event of an actual number of deaths that has occurred in ahistorical dataset for a known group of pension scheme member lives; and

FIG. 31 shows a plot of the probability density function produced by aMonte Carlo model and a Poisson distribution model for an expectednumber of deaths for a known group of pension scheme member lives.

DETAILED DESCRIPTION OF THE INVENTION

As shown in FIG. 1, the pensions defeasance products will be issued inboth securities (1) and derivatives (2) form. For this purpose, both maybe issued from a single entity, or two distinct issuing entities mayexist. The defeasance products will be issued as cash securities (S)under the a Pensions Defeasance Master Trust, a cell company or a masterissuing company and silo structure (PDMT) and in derivative form (D)from the PDMT, or a separate Pension Derivative Products Company (PDPC).

A Master Trust, cell company or master company and silo (MT) arestructures often used in the asset backed securities market e.g. creditcard issuers. The PDMT may comprise known capital markets structures.

At least one Pensions Sub-Trust, cell or silo (PST) is provided beneaththe PDMT. The capital structure of the PST's combines threads oftechnology of known capital markets structures.

Similarly, the PDPC uses technology found in Derivative ProductsCompanies (DPC).

In each case technology in accordance with the invention is used inorder to facilitate the assumption of risks peculiar to the pensionsmarket, such as longevity and in so doing creates an entirely newsolution to pensions risk transfer, together with new asset classes inthe capital markets and as such represents a new business application.

Similarly, the risk management system, shown in FIG. 4 is based on acombination of existing capital markets and pensions market systemstechnology, which has been integrated into an entirely proprietaryreporting framework. The risk management system provides a systemcapability which did not previously exist and which allowssecuritization technology to be used to achieve comprehensive defeasanceand risk transfer from the pensions market to the debt capital markets.

Referring now to FIG. 2, a schematic of the entities involved in theissue and trading of pension defeasance securities will now bedescribed. Securities will be issued under a PDMT, which will be—forexample—a AAA/Aaa rated program, rated by leading rating agencies, suchas Standard & Poor's, Fitch and Moody's rating agencies. The PDMT willbe established as a special purpose entity based in, for example, Jerseyor another appropriate location.

Beneath the PDMT, will sit a family of PST's, which will issue financialinstruments according to embodiments of the invention in the form ofPension Defeasance Trust Certificates (PDTCs), bonds, notes or othersecurities (aka. Pension Defeasance Securities) to investors (such as,pensions schemes, insurance companies and derivative counterparties,such as investment banks) seeking to immunize their exposure to pensionsrisk.

The PDTC's will be issued under a global issuance program (under whichdedicated targeted programs can exist for specific jurisdictions wherelocal securities law requires), which will be listed on at least onemajor international stock exchange. The certificates will be open tosubscription through a group of appointed dealers and will also be opento reverse enquiry from dealers outside the program, under “dealer forthe day” arrangements similar to those which typically exist on mediumterm note programs.

One of the key features of the program is its flexibility to issuespecific tranches of PDTC's (where appropriate out of dedicated PST's)which substantially meet the exact risk profile of the investor, so asto ensure complete economic defeasance and therefore complete transferof risk. This means that the permutation of options available toinvestors under the program is almost unlimited, providing that theexposures are capable of being hedged, or managed under the criteriaagreed with the rating agencies for the preservation of the ratings ofthe PDMT or PST's senior obligations. The criteria agreed with a ratingsagency for the preservation of a rating are be set out in a RiskManagement Manual and/or an Operations Manual associated with financialinstruments issued under the program in the form of PDTCs, and saidfinancial instruments will be operated in accordance with the RiskManagement Manual and/or an Operations Manual such that the ratingsagency rating is achieved and maintained.

PDTC's issued under the PDMT will generally carry a stand alone AAA/Aaarating, thus putting them on a par with the obligations of the highestrated governments and corporate entities and above the credit of manysovereign entities and most banks and insurance companies. However,where required, the facility exists within the program structure tooverlay a third party AAA/Aaa guarantee, typically provided by amonoline insurer, or similar entity, thereby adding further to theintegrity of the covenant. The resulting instrument will be issued as aGuaranteed Pension Defeasance Trust Certificate (GPDTC).

Each sub-Trust will be dedicated to a specific class of risk. Forexample, PST1 might issue PDTC's where the payments due to investors arelinked to a specific longevity index, such as the Continuous MortalityIndex (CMI) or Lifemetrics. Investors in this class of PDTC's wouldtherefore receive payments on their PDTC's which mirrored theperformance of the appropriate index. That is, if longevity improves,meaning that people are living longer, the payment flow on the PDTC'swill extend accordingly. Investors in such a tranche, might include, forexample, pensions schemes looking to partially hedge their liability ata cheaper cost by purchasing a generic rather than bespoke hedginginstrument, leaving them to manage the basis risk between the index andthe actual performance of their scheme. This type of instrument might bechosen by an insurance company or derivative counterparty such as aninvestment bank with the capital and technical expertise to manage theresulting basis risk.

PST2 might issue PDTC's indexed to the performance of an individualcompany's pension scheme (for example, the pension scheme of a major UKcompany). The performance of this bespoke tranche will therefore mirrorthe performance of the particular scheme. That is, the cash flows on thePDTC's will reflect improvements in longevity, track inflation ifappropriate, and reflect in aggregate all of the events impacting theportfolio of individual pensions of which the scheme comprises (such asspouse and dependant obligations election to take lump sums onretirement, transfers out of the scheme, etc.). The investors in thesePDTC's might be, for example, the pension scheme itself to hedge itspension liability, or derivative counterparties, which have exposure tothat particular pension scheme. By investing in this product theinvestor would be immunized from exposure to the relevant pensionsscheme and would have no basis risk to manage. This is because there is,a very high degree of correlation between the mortality of the referencepopulation used to determine the cash flows of the PDTC's and themortality experienced by the particular pensions scheme.

PST3 might issue a tranche of defined term PDTC's, which instead ofbeing linked to a generic index, or to the underlying obligations of ascheme (for example, another major British company) until the death ofits last member, would provide a hedge for a scheme's pension liabilityfor a specific period of time. For example, the payments on the PDTC'scould be set to reflect the experience of the pensions scheme in termsof meeting all of its payment obligations for a ten year period. ThePDTC's obligations might also incorporate the obligation to deliver alump sum on maturity equal to any deficit (under IAS 19 or whichever isthen the appropriate accounting standard) which may exist between thepensions scheme's assets and liabilities on the maturity date. Inaddition the PDTC's might also include, for good measure, the obligationto cover the cost to the scheme of a credit default, or failure of thesponsor at any time during the life of the PDTC's. An investorpurchasing these securities would therefore have defeased or immunizedthe longevity risk and all other pensions scheme exposures for therelevant pensions scheme for a period of 10 years; have ensured that atthe end of 10 years the relevant pensions scheme has no deficit (if thescheme has a deficit on the date of the PDTC's issuance this will ineffect provide scheme deficit financing over a 10 year time horizon);and protection against a credit default by the pensions scheme'ssponsoring employer.

The PDTC's will generally represent the A class and senior securedinterests of each PST, as shown in FIG. 4. These ratings will beachieved by supporting the financial instrument with an amount of riskcapital. The amount of risk capital to be held may be determined byquantifying the risk exposure of the financial instrument.

Risk capital may be raised by the PDMT which may act as a capitalcompany and the amount of risk capital needed to support each PST may bepassed on to that PST. The amount of risk capital passed to the PST maybe determined to be an amount sufficient to support the PST's exposureto an amount of longevity risk and an amount of asset risk. In issuingthe PDTC's, each PST may receive an investment amount from investors inthe PDTC's and the PSTs may use the investment amounts to invest inassets to fund the payment amounts to be paid on the PDTC's.

Another possible arrangement is one in which the PST acts as an issuingentity and issues the PDTC's to investors in return for receiving anamount of investment, the PST then transfers the investment amount anasset holding entity also within the cell. It is then the asset holdingentity that invests in assets to fund the payment amounts to be paid onthe PDTC's, the asset holding entity transferring to the PST amountsmatching the payment amounts on the PDTC's issued thereby. In thisarrangement both the PST and the asset holding entity are supported byan amount of risk capital raised and passed on by the PDMT.

The exposure of the financial instrument to longevity risk may bequantified in accordance with methods of aspects of the presentinvention. To raise this risk capital the obligations of the PDTCs willbe supported by the issuance of stratified subordinated classes ofjunior financial instruments in the form of Pension Defeasance CapitalCertificates (PDCC), bonds, notes or other securities, which will berated according to their priority in the sequential payment waterfalland further underpinned by further subordinated unrated PensionsDefeasance Equity, or Capital, certificates, bonds, notes or othersecurities (PDE). The amount of subordinated debt and equity raised byissuance will make up the risk capital supporting the PDTC.

The risks apportioned to the PDCC's and the PDE, will together encompassall of the exposures of the specific PST for which they provideenhancement. These may include exposure to longevity, inflation,interest rates, currency, credit, equity, property and alternativeinvestments. The specific exposures borne by investors in each class ofPDCC's and PDE, may be tiered simply in terms of seniority, in whichcase income of the PST after payment of its fees, expenses, seniorobligations and any requirement for retention under conditions agreedwith the rating agencies, will be paid out according to a prioritywaterfall. Alternatively, the individual classes of PDCC's and PDE's maybe specifically linked to the performance of a single class of risk orspecified grouped exposures, i.e. just longevity, or longevity andinflation, but no other exposure within the portfolio.

With regard to derivatives, the derivative products will largely mirrorthe aforementioned securities products already described, but will beissued in the form of derivative contracts, including total returnswaps, futures contracts and contracts for differences and may be issuedthrough the PDMT, or through a PDPC.

FIG. 3 illustrates a data processing system 100 for use in carrying outmethods in accordance with the invention. At a local site there arepersonal computers 101, 102 and 103, which are interfaced to a localnetwork 104, and a local server 105 which is also interfaced to thelocal network 104. Data can be stored on the local server 105 and/or thepersonal computers 101, 102, 103. Data processing can be carried out bythe local server 105 and/or the personal computers 101, 102, 103. Thelocal server 105 and/or the personal computers 101, 102, 103 may beconfigured by software to carry out the steps of methods in accordancewith the invention. The local network 104 is provided with an interface106 to a wide area network 107, so that the local server 105 and thepersonal computers 101, 102, 103 communicate with the wide area network.Remote servers 108 and 109 are also connected to the wide area network,so that data held by the remote servers can be made available to localserver 105 and/or the personal computers 101, 102, 103. The remoteservers can receive data from data feeds 110 and 111 also connected tothe wide area network 107, which provide data such as mortalitystatistics, pension fund statistics and so forth. This basic data isprocessed by the remote servers 108 and 109 so as to provide data whichis used by the local server 105 and the personal computers 101, 102, 103in carrying out the methods in accordance with the invention.

Risk Management System

FIG. 4 shows the operation of the risk management system of the presentinvention and is set out in the form of a flow chart showing thetransfer of data between the different modules making up the riskmanagement system. The risk management system is an integral componentof the pensions defeasance system of the present invention is, as shownin FIG. 4, can be notionally divided into five operational layers: a rawdata input layer; a data input layer; an asset and liability enginelayer; a product/trading platform layer; and a reporting layer. Modulesof the system shown with a solid outline in FIG. 4 denotes a new moduledeveloped to make up the risk management system. Those modules shown inFIG. 4 with a dashed outline denote a module that existed in some formbefore the development of the risk management system but which has beenimproved before incorporation in the risk management system. Thosemodules shown with dash-dot-dash outline in FIG. 4 denote a module thatexisted in the form in which it has been incorporated before thedevelopment of the risk management system.

These operational layers of the system and the modules of the systemthat make up each layer will now be described in turn. For eachoperational layer of the system, the current situation of the operationof a pension scheme before the development of the present operation willfirst be compared with the operation of that layer of the riskmanagement system of the present invention.

Raw Data Input Layer

This layer relates to the collection, analysis and availability of alldata required to be input to the risk management system of the presentinvention.

Currently, pension scheme data is collected on ad hoc basis and is ofvery poor quality; population and industry data are analysed only byacademics but basis risk is not focussed on; and market data is nottransparent to non-participants.

In accordance with the risk management system underlying the presentinvention, a rigorous data collection process allows the drilling downto the most detailed level of analysis across all elements of the rawdata inputs.

The raw data input layer comprises: a pension scheme data module; apopulation and industry mortality experience data module; and a marketdata module.

In the pension scheme data module, pension scheme data is currentlyrecorded by third party administrators or in-house administrators ofpension schemes and the quality of data varies and is generallyextremely poor. Data cleaning for a bulk buy-out exercise typicallytakes between 6 months to 2 years to complete. In the risk managementsystem of the invention, this module creates a standard data protocol toprovide a link between the pension scheme data and the trading platformand ensures minimum data quality standards are met through standard dataquality control and checks. This module must be linked to the systems ofany preceding third party administrators of the pension scheme.

In the population and industry mortality experience data module, a largesample of data of a suitable reference population is required to beinput to the risk management system in order to make credible forecastsof mortality and longevity trends over time. Currently the only sourcesof data sufficient in size in the UK are (i) population data from theONS and (ii) insured population data collected by the CMI. In the riskmanagement system of the invention, this module provides a consistentway to extract the relevant data for projection of longevity trends.This module is linked to have access to the mortality data of thesuitable reference population.

In the market data module, data from the relevant markets that isrequired by the risk management system is collected. The market datamodule may collect all relevant swap curves sourced from marketcounterparties on a daily basis, all pricing information required tobuild proprietary pricing curves, all pricing information required torun a daily mark to market on all assets contained within the pensionscheme, and all pricing information required to create hedging exposuremaps. In the risk management system, this module provides access to allrequired pricing inputs and all swap curves required for valuing pensionportfolios on a daily basis and also stores and collates relevantpricing information to allow a daily mark to market on all assetscontained within a pension scheme. This module requires access toclosing market data on a daily basis.

Data Manipulation Layer

The data manipulation layer relates to the ‘cleaning’ andstandardisation of data input from the raw data input layer and to theso that it meets the operation parameters of the risk management systemof the invention and to valuation of hedging pension portfolio assetsdependent on market data.

Currently, data cleaning specialists operate on a project/contractbasis, but no standard data protocols exist, mortality analysis is ledby academic and industry benchmarks do not exist, and hedging assets arebespoke products sold by Investment Banks.

In accordance with the risk management system underlying the presentinvention, standardised data protocols and transparent mortalityassessments will lead to a market standard and an open-architecturehedging strategy.

The data manipulation layer comprises: a pension data cleaning andstandard data formatting tools module, a longevity assumption settingtool module, and a hedging asset valuation tool module.

In the pension data cleaning and standard data formatting tools module,pension scheme data extracted and cleaned to convert it into a standardprotocol that meets the operational parameters of the risk managementsystem. Currently, data that is currently extracted from pensionadministrators is not standardised. In addition to poor data quality,there is currently no motivation for Trustees or Sponsors to see datacleaning as priority. This means data cleaning is currently an ad-hocprocess and mostly applied in a wind-up situation, thus many corporatesponsors do not have an accurate assessment of the full liabilityexposure of the pension scheme. In the risk management system, thismodule provides a standard data transfer protocol between third partypension scheme administrator systems and the systems risk managementsystem. It also provides standardised procedures for data cleaning andon-going data maintenance. It provides an ability to source andincorporate additional information to improve the quality of dataprovided by each scheme. It ensures a minimum data quality on which a‘clean’ pricing can be achieved. It also creates a “market standard” forinformation content and quality required from pension schemes andtrustees. This module is linked to the pension scheme data module whichinputs the collected pension scheme data.

In the longevity assumption setting tool module, mortality analysis isconducted applying models to identify trends in mortality to valuepension liabilities and enable quantification of the longevity exposureof the pension scheme and determine a risk capital requirement.Currently, longevity trends are generally analysed in a number ofacademic papers but there is no common approach adopted by themarket/industry. In the risk management system, this module incorporatesthe leading mortality/longevity models in a consistent and transparentmanner to provide longevity assumptions to value pension liabilities andfor determining capital adequacy requirements for rating purposes. Thismodule contains functionality allowing scheme specific mortalityadjustments based on sex, age, size of pension and socio-economicfactors through post code analysis. It extracts longevity projections(both mean estimates and tail scenarios) from reference population data,for example, CMI data. This module is also capable of determiningadjustments for mortality rates linked to socio-economic groupings andspecific pension scheme profiles. This module is linked to the pensionscheme data module and the population and industry mortality experiencedata module. It reads mortality experience data from the referencepopulation data such as the ONS and CMI sources and reads the pensionscheme data and builds up scheme specific mortality experience overtime.

In the hedging asset valuation tool module, the assets of the pensionscheme portfolio to be hedged are valued in accordance the market data.The current approach to hedging a pension portfolio is by providinghigh-level duration information (e.g. PV01) followed by raw data to theproviders of the derivative instruments without a standard approach. Anumber of providers offer an investment solution comprising a range offunds which approximate the underlying investment risk profile of apension scheme, for example, LDI providers. Both approaches require theinvolvement of fund managers or investment consultants as “middle men”.In the risk management system, this module creates exposure maps tofacilitate risk management of the portfolio and the system also includesa pricing module that uses appropriate proprietary swap curves inpricing a proposed transaction. This module is linked to the market datamodule and requires access, for example, to Bloomberg mid-market screensand the relevant ‘pricing grids’ from swap counterparties to buildappropriate proprietary pricing curves.

Asset and Liability Engine Layer

The asset and liability engine layer relates to the projection of thepension scheme cash flows and to the calculation of the risk capitalrequired in order to achieve a rating from a rating agency.

Currently, a range of liability cash flow models exist but they requireto be individually adapted so that they are bespoke for each pensionscheme/client and they require actuarial knowledge and programmingexpertise. A rating agency capital projection framework and model do notcurrently exist.

In accordance with the risk management system underlying the presentinvention, a robust cash flow and capital projection system is achievedwith minimum tailoring to each pension scheme/client and a rating agencycapital projection framework is provided.

The asset and liability engine layer comprises: a cash flow projectionmodel module; a longevity capital model module; and an asset platformmodule.

In the cash flow projection model module, expected cash flows of thepension scheme are projected. Currently, a range of cash flow projectionmodels exist that could be used to model pensions and annuity business.However, the models require significant modification for each pensionscheme and require both programming and actuarial expertise. In the riskmanagement system, this module imports pension scheme information forboth benefit entitlements and member data using a standardised approachthrough a customised database front-end. The standardised pension schemedata protocol is used. Each “slice” of a member's pension entitlementsis modelled using a flexible approach adaptable across multiplejurisdictions and geographic regions. The module projects expected cashflows using assumptions linked to other risk management modules withinthe wider system and allows valuation of all pension risks, such as,transfer-out value, cash commutation, orphan benefits, etc, in additionto longevity and market risks. This module is linked to the precedingpension data cleaning and standard data formatting tools module fromwhich it imports pension scheme data and also this module has access tobest estimate assumptions from the longevity assumption setting toolmodule.

In the longevity capital model module, the longevity risk exposure ofthe pension scheme is measured and quantified. Currently, pensionschemes are not required to capitalise for longevity risk. Lifeinsurance companies currently use simplistic capital calculations forlongevity risk and these are mainly scenario driven. Rating agencyapproved longevity capital models do not currently exist. In the riskmanagement system, this unique and proprietary module contains themethodology and process for quantifying longevity risk within a pensionscheme for the purposes of obtaining short and long term debt ratings upto and including AAA/Aaa ratings from ratings agencies. This moduleincludes a rating agency approved capital model that allocates andprojects the longevity capital requirements of a pension scheme oneither a deterministic or stochastic basis. This module is linked to thelongevity assumption setting tool module from which the best estimatemortality assumptions are imported and is also linked to the cash flowprojection model.

In the asset platform module, cash flows for all asset classes held bythe pension portfolio are projected. The existing asset platforms usedby insurance companies, pension schemes and pensions consultants arelimited to modelling a broad representative asset portfolio and do notincluded granularity at individual stock level. Asset allocation andportfolio decisions are currently modelled on a high-level, for example,an X % equity proportion. In the risk management system, the assetplatform module models and projects cash flows for all asset classes. Itmanages and records trading activity and creates curves for pricing,hedging and risk management. It allows direct linking of assets andliabilities allowing analysis and hedging on a portfolio or individualbasis and it contains functionality allowing sensitivityanalysis/management of 01 exposure. The asset platform module is linkedto the market data module from which it reads in market data on a dailybasis and to the total portfolio management system module from which isaccesses asset and liability portfolio information.

Product/Trading Platform Layer

The product/trading platform layer relates to the trading of the capitalmarkets products associated with the risk management system which enableto securitization of pension liabilities.

Current pension risk transfers are assessed on a case-by-case basis by ateam of actuarial specialists. A single pension projection and capitalmarket trading platform does not currently exist and pension liabilitiesare not directly linked to asset trading platform.

In accordance with the risk management system underlying the presentinvention, a single platform allowing a combination of pensionprojections, trading of capital market products and direct linking ofasset and liability portfolios is achieved.

The product/trading platform layer comprises a total portfoliomanagement system module.

In the total portfolio management system module, the capital marketsproducts, assets, liabilities and pension projections underlying thesecuritization of a pension scheme in accordance with the presentinvention are combined on a single platform. The market currently allowspensions risk to be transferred only to an insurance company through abulk annuity exercise. The bulk annuity exercise is assessed and pricedon a case-by-case by a team of actuarial specialists and the assumptionsand the details of the transaction are not transparent to the pensiontrustees nor wider public. A solution that allows risk transfer ofpensions liability risks to the capital markets does not currently existand hence a single pension projection and capital market tradingplatform does not exist. In the risk management system, the totalportfolio management system module provides a single platform allowing acombination of the following things. Dynamic pension liability cash flowprojections including: analysing cost and liquidity impact of excisingmember options; analysis of hedging requirements/costs/strategies;comparison of actual versus expected; assessment of correlated riskexposure e.g. longevity improvement event compounded with a rise ininflation. Detailed analysis of pension slices. Asset modelling and cashflow projections. A trading system for all classes of capital marketproducts. Analysis and trading of derivative products. Production oflinked asset and liability portfolios. The production of exposure mapsfor risk management. Micro hedging of individual asset and liabilitycash flows. Allocation of capital through rating agency approved assetand liability models (both stochastic and deterministic). All aspects ofliquidity management including projecting tracking and analysing cashflows generated by both assets and liabilities. Ability to price andmark to market all assets and liabilities. Daily, comprehensive andtransparent reporting. The total portfolio management system module islinked to the cash flow projection model module to allow it to readmember level cash flows, to the longevity capital model module to allowit to read capital requirements, and to the asset platform module toallow it to read both asset and liability data to create a portfoliobased approach.

Reporting Layer

The reporting layer relates to the reporting of the outputs of theproduct/trading platform layer to various stakeholders.

Pension valuation reports are currently in the form of: (i) triennialactuarial valuations (ii) accounting values (iii) bulk annuity quotesfrom insurers. There is currently no disclosure of mortality assumptionsfrom any of these reporting sources, and, as a result, a total lack oftransparency in current pensions reporting.

With regard to reporting for rating agencies, to date Rating agenciesparticipation in the pension scheme risk transfer has been limited toproviding ratings for derivative swap counterparties (principallyinterest rate and inflation swaps and single cohort longevity bonds).The rating agencies have not previously provided ratings for any productwhich achieves comprehensive risk transfer such as is achieved by theembodiments of the present invention which are capable of immunisingmultiple facets of pension portfolio risk including actual longevityexperience, inflation, early retirement, spouse and dependant pensionentitlement election to take lump sums on retirement and transfers outof the scheme. Therefore no reporting to rating agencies is currentlyprovided.

With regard to reporting for capital investors, capital market investorsare currently not directly participating in pensions risk transfersolutions and therefore no reporting to capital investors is currentlyprovided.

With regard to reporting for pension scheme trustees, the most accuratepension valuation reports are currently carried out once every threeyears and they take a minimum of 6 months to complete. This meanssponsors do not currently have up-to-date information on their pensionobligations.

With regard to reporting for internal purposes, current internal riskassessment includes accounting valuations (e.g. FRS 19) and ALM models.FRS 19 is widely recognised by experts as inadequate reflection of thetrue underlying risk. ALM models are used to manage high-level riskdecisions such as equity mix.

In accordance with the risk management system underlying the presentinvention, comprehensive, transparent, web based reporting to multiplestakeholders and potentially wider public is achieved. The reportinglayer comprises: a rating agency reporting module; a capital investorreporting module; a pension scheme trustee/employer reporting module;and an internal risk reporting module. In the risk management system,the reporting layer modules achieve this reporting to such stakeholdersincluding sponsors, investors, market counterparties, rating agenciesand potentially the wider public. Reporting in the risk managementsystem includes asset/portfolio reporting, in which: asset rating/assetclass/issuer concentration/geographic concentration reports areproduced; capital management and allocation reports are produced;liquidity reports including daily cash flow projections are produced;Hedging/interest rate/inflation sensitivity analysis and reports areproduced; cash management reports are produced; and asset and liabilityprofile reports are produced. Reporting in the risk management systemalso includes liability reporting, in which: monthly cash flowprojection reports are produced; reports of major valuation assumptions,for example of mortality levels/trends, are produced; key member profilestatistics reports are produced; IRR and payback period of capitalinvestment reports are produced; actual versus expected analysis reportsare produced; and capital risk exposure and expected loss analysisreports are produced.

At the core of the parameters agreed with the rating agencies to securethe appropriate debt ratings for the senior and junior debt obligationsof the PST are the capital projection models, which evaluate risk (inthe form of, for example, expected loss) within the portfolio todetermine how much capital is required to support the issuance of seniordebt obligations. These are proprietary models operate within the riskmanagement system and lie at the heart of the rating methodology.

The models shown in FIG. 4 can be run deterministically orstochastically and are run daily to measure rating agency compliance.The capital models collectively determine how much risk capital isrequired each day to comply with the rating agency requirements andcapital is measured as a combination of subordinated debt, equity. Thismay also include the excess spread i.e. the net spread between cost offunding (collectively senior and junior debt) and the income from theinvestment portfolio.

Issuing and administering a longevity financial instrument The processof marketing and selling the PDTC's will vary according to theircharacteristics. Some tranches will be originated through reverseenquiry, where the investor(s) or dealers will define the risks whichthey are seeking to hedge though the purchase of the certificates.Alternatively, tranches may be structured on the basis of establishedparameters and indices and offered to the wider market.

Where the issuance is by reverse enquiry, the execution timetable willlikely include an extensive due diligence process, which will involvethe collection of relevant data (potentially up to and including data onall of the members of the scheme) needed to price the offering. Wherethe issuance is to be linked to a specific pension scheme the memberdata will need to be obtained from the scheme or from third partyadministrators, collated and “scrubbed” or “cleaned” to meet theparameter requirements of the risk management system. Also, all of thepension rules (both the pension scheme and regulatory rules) and anyother relevant parameters will need to be modeled within the riskmanagement system, so that a defined liability can be determined, albeitwith variable parameters.

Assuming that the pricing of the PDTC's meets the targets of theinvestor(s), the investor(s) will then subscribe for a dedicated trancheof certificates.

Within the PST a complex process managed and monitored by the riskmanagement system then begins, which will now be described. Upon receiptof funds subscribed to purchase the PDTC's, and the issuance ofappropriate tranches of PDCC's and PDE to capital investors, inaccordance with the requirements of the capital model, the PST willimmediately commence the process of hedging and managing the complexliability which it has acquired.

This will include writing inflation and interest rate swaps with marketcounterparties to translate the liability (which can be thought of as along sequence of zero coupon obligations, albeit a sequence which canexpand, extend or contract) into a floating rate LIBOR based cash flow,to remove inflation and interest rate risk from the portfolio. For othertypes of liability, such as index based transactions, term longevityhedging, deficit elimination or sponsor default protection, other typesof primary hedging may be used to enable the PST to be managed withinthe agreed ratings criteria. Where the underlying risks cannot be fullyhedged, the PDMT and the PST's will hold sufficient additional capitalaccording to the levels determined by its deterministic or stochasticcapital models to satisfy the rating agencies that the risks are coveredto the appropriate level to ensure that all of its securities orderivatives obligations can be met on a timely basis.

The subscription funds will then be invested in LIBOR based investmentproducts. The investment process will initially be in cash deposits andother short term cash instruments. However, because the liabilities ofPST's will typically be long term in nature, reflecting the mortalityexperience of the pensions scheme, the investment process will bedynamic and designed to achieve economic defeasance of the PST'sliabilities over the medium to long term.

The PST will operate under investment parameters agreed with the ratingagencies, which will allow it to extend its investment profile from cashand fixed income investments, right through to equities, property andalternative investments. The PST's investment portfolio will thereforebe determined on a dynamic basis, according to available assets, marketconditions (pricing), available capital, cost of capital and liquidityrequirements, all measured within the risk management system against acapital model agreed with the rating agencies.

The risk management system, shown in FIG. 4, will enable theadministration of a complex set of monitoring and management tasks whichwill help ensure that the PST always remains fully compliant with itsrating obligations and meets its obligations to investors on a timelybasis. The list of daily tasks includes:

-   -   Running the asset and liability capital models to ensure capital        compliance;    -   Marking all assets and liabilities to market;    -   Measuring the 01 portfolio volatility and rebalancing        derivatives and other hedges to take account of changes in the        investment portfolio and the profile of the liability to ensure        compliance with agreed sensitivity limits;    -   Running both a short term (1 year) and long term (to the final        date of the longest liability) net cumulative outflow test to        ensure the PST will always have liquidity to meet its payment        obligations;    -   Measuring all sector concentrations, such as geography,        industry, sector and country, to ensure compliance with rating        agency diversification tests;    -   Monitoring scheme data such as deaths (actual versus projected),        spouses/dependants, withdrawals etc. to ensure that the profile        of the PST's liability always accurately reflects changes to the        scheme's pensions liabilities

In addition to these daily reporting tasks, which in total will requirethe production of a large number, for example 150, daily reports therewill be an extensive mid and back office administration functionrequired to ensure compliance of the PST with all of its obligations toinvestors, rating agencies and Stock Exchange(s). These administrativefunctions will encompass, rating agency reporting, accounting,securities and derivative settlements, pricing, trustee functions,custody and paying agency and cash management.

Using this system, a pension scheme will be able to purchase aninvestment, or enter into a derivative contract, with the capability toprecisely mirror the liability profile of a part, or all of its pensionobligations. In doing so, the sponsoring employer and the Trustees ofthe scheme will know that they have fully transferred the embedded risksof that part of the scheme which has been hedged, to investors in thecapital markets. Depending on the specification of the securities orderivative contracts in which the scheme invests, this means that someof, or the combined risks of longevity, inflation, interest rates,currency, credit, equity, property and alternative investments will havebeen removed from the scheme for the life of the investment. TheTrustees will be safe in the knowledge that the scheme's obligationswill in future be met from the income received from the scheme'sinvestment in PTC's or D's to a AAA/Aaa standard, or such lower ratingas the scheme specifies, and the sponsoring employer will have nofurther exposure to the pensions deficit volatility that a pensionscheme can impose upon its balance sheet.

Case Study of Implementation of the Invention

The following is a potential case study as an example of how aspects ofthe present invention may be implemented. This case study looks at theapplication of aspects of the present invention as a solution for amature UK pension fund. To provide perspective, the case study alsolooks at the alternative options available, based upon the UK PensionRegulator's list of risk transfer options available to UK pension funds,published in December, 2006 (reprinted below). The case study also looksat the impact of the reporting and risk management systems.

In accordance with aspects of the present invention, for the first timepension schemes are able to purchase investment securities, or enterinto derivative contracts, the cash flows of which will accuratelyreflect the liability profile of their obligations to pensioners.

In doing so the sponsoring employer of the pension scheme and itstrustees will be able to transfer the embedded risks (such as longevity,inflation, interest rates, currency, credit, equity, property andexposure to alternative asset classes) to the capital markets andthereby defease the scheme's pensions liability.

The case study concerns a hypothetical corporation, ABC Airways. ABCAirways (ABC) is a formerly nationalized European airline, which wasprivatized in the 1980's with a large legacy pension scheme. The totalestimated size of its pension scheme obligations, including pensioners,active employees and deferred members (former employees who have not yetretired), is £15 billion. ABC's current market capitalization is £5billion.

Changes in pensions legislation, combined with recent accounting rulechanges have forced the pension scheme to the top of ABC's managementagenda. Having formerly been regarded as a contingent liability whichwas not recorded or recognized in ABC's accounts, management now has tocontend with the following pension related issues:

-   -   Any deficit between the estimated pensions liability (measured        under FRS 17 and IAS 19 on the basis of AA rated bond yields)        and the value of the scheme's investment assets, much of which        is invested in equities, must now be recorded as a debt to third        parties in the company's balance sheet.    -   ABC's current deficit, based on its most recent actuarial        valuation, which was carried out in 2005, is £2 billion. The        mortality assumptions used as the basis for this estimate of the        deficit are not published.    -   The UK Pensions Regulator has expressed concern at the size of        the deficit and requires the company to show that it will be        able to reduce the deficit to zero within 10 years. The company        plans to achieve this by a combination of special contributions        and transferring certain property assets to the scheme.    -   Due to the size of the deficit, the Regulator has also exercised        its powers to require ABC to suspend all forms of distributions        to its shareholders, including dividend payments, until there        has been a substantial reduction in the size of the deficit.

The credit rating agencies, seeing the pensions deficit as an obligationto third parties and part of ABC's debt, have reduced its credit ratingto the non-investment grade level of BB.

-   -   The cost and credit availability implications of this downgrade        are a serious concern for ABC, which will need to start        purchasing new aircraft from 2008 if it is to maintain the        quality of its fleet.    -   In an effort to reduce the burden of the scheme on the company,        senior management has announced that it intends to negotiate        with its pension trustees and the unions to seek changes in the        schemes benefits. However, the unions which are very strong in        the airline industry, have indicated that any initiative to        reduce pension benefits, or increase the retirement age will        lead to industrial action.    -   Despite the difficulties which many others in the industry have        faced, ABC has a strong management team and despite substantial        increases in fuel costs, is actually on track to meet its target        of achieving an operating margin of 10% within the next two        years.    -   The company is also likely to be one of the major players in the        anticipated consolidation of the airline industry, which is        likely to follow from the recently agreed “Open Skies” policy.    -   As with other former nationalized airlines, which are similarly        burdened with legacy pension schemes, ABC's share price reflects        the impact of the pensions scheme—equity analysts have described        the company as a large pension scheme run by a small airline—and        the disparity between the multiple of EBITDA on which ABC trades        compared to the new low-cost carriers, which do not have a        similar pensions burden is striking.    -   The fact is that despite its success at running its core        business, ABC is a very complex story for the equity markets to        understand as it can no longer be valued as a straight airline        stock due to the balance sheet impact of its volatile pensions        deficit.    -   To determine an accurate valuation of the company, analysts        would also need to reflect the volatility of the £13 billion        pension asset portfolio, much of which is invested in        equities—on this basis, ABC is arguably a 3× leveraged equity        play.    -   On the other hand ABC's pensions liability is valued on the        basis of corporate bond yields and therefore its share price        also needs to reflect volatility in bond yields.    -   Unfortunately, this is just too complicated for the poor share        analysts who do not have the transparency of information to make        these calculations (the fact that the pensions liability is only        re-calculated every three years itself makes a mockery of any        attempt to conduct a marked to market valuation) or the tools to        carry out this analysis and so as with any business which they        do not fully understand, they mark the stock down.    -   Starved of dividends, with a significantly underperforming share        price, ABC's investors are frustrated and confused by the impact        that the pension scheme is having upon what by all measures is        otherwise a very successful company.    -   Beyond the existing investors, potential bidders from the        private equity markets are also frustrated by the uncertainty        resulting from the pensions issue. Thus, a company that ought to        be at the centre of attention in an industry that is likely to        experience intense consolidation, continues to under-perform.

ABC's Options Using Only Conventional Solutions

-   -   Based on the list of options available to pensions schemes to        achieve risk transfer, as defined by the UK Pensions Regulator        in December, 2006 (reprinted below), ABC's management have only        a limited menu of solutions to their pension problems, none of        which is sourced from the capital markets.    -   They could close the scheme to new members. This would be deeply        unpopular with existing employees and management recognize the        value of the scheme as a human resources tool.    -   Alternatively, they could retain the scheme, but reduce the        benefits and increase the age of retirement. This is also seen        as a very unpopular move and while some progress might be made        on this front, it is unlikely to be enough to eliminate the        deficit and bring the cost of the scheme to a manageable level.    -   Neither of these strategies in itself will deal with the        fundamental problem of the volatility of the deficit or indeed        surplus.    -   To complicate matters further, ABC's pensions liabilities are        spread over a number of schemes, two of which are closed to new        members. While the company's overall deficit across its schemes        is £2 billion, the two closed schemes are both slightly in        surplus.    -   ABC therefore finds itself in the frustrating position that were        market conditions to become favourable (the ideal combination        would be rising share prices and rising interest rates), would        on the one hand benefit the schemes in deficit, while on the        other there would be no way to claw back the growing surpluses        in the two closed schemes, due to the asymmetry under which        deficits in both open and closed schemes sit on the sponsor's        balance sheet, while surpluses in closed schemes belong to the        scheme members and cannot be clawed back by the sponsor.    -   ABC pension trustees have taken advice from an investment bank,        which having analyzed the funds, concluded that they were        exposed to three types of risk: equities, interest rates and        inflation—for some reason they ignored longevity, possibly        because they had no solution to offer (see, for example, the W.H        Smith Case Study published on the UK Pension Regulator's        website, reprinted below).    -   The investment bank's proposal was to implement a 95% swap        overlay liability driven investment strategy (“LDI”), using        indexation and interest swaps, combined with a 5% investment in        equity options.    -   However, the trustee's investment consultant pointed out that        while this strategy would protect against further ballooning of        the liability caused by a further fall in bond yields, the        analysis ignored the scheme's exposure to longevity, for which        the investment bank had no product solution. Thus the scheme        would remain exposed to the uncertainty of its member's        mortality and thus uncertainty about its ability to meet its        future obligations.    -   Two other investment banks proposed derivative solutions based        on generic population longevity indices. However, the proposals        would have left the pension scheme with considerable basis risk        (the differential between the index on which the derivatives        would have been based and the likely longevity experience of its        own pensioner population) and there was also some doubt about        the banks ability to execute the transactions.    -   Having rejected the LDI strategy and the indexed derivatives        because of their inability to deal with the schemes specific        exposures to longevity, the trustees looked at bulk annuity        purchased from an insurance company as a potential solution.    -   Bulk annuity would certainly provide a full risk transfer of the        scheme's obligations to its pensioners, but there were issues of        scale, cost and the quality of the covenant.    -   In terms of scale, ABC's total pension's liabilities of £15        billion were considered way beyond the present capacity of the        market, which has typically operated with an annual volume of        around £2.5 billion. New entrants have joined the market        recently, but even with the new capital which they have brought        to the market, the scale of ABC's requirement would be well        beyond the market's current capability.    -   Another negative factor when considering bulk annuity was cost.        Bulk annuity utilizes regulated insurance company balance sheet        capacity based on the equity capital of the insurance company        and is therefore an expensive product. It is also a far from        transparent product and ABC's trustees were troubled by the fact        that neither their investment consultant, nor employee benefit        consultant were really able to explain the basis on which the        product was priced.    -   Notwithstanding the cost and lack of scalability for a total        pensions scheme defeasance, ABC did look at bulk annuity as a        partial defeasance i.e. as an investment asset of the scheme,        rather than as a full legal and economic transfer of the        scheme's obligations to its members. However, they decided        against this option eventually on the advice of their lawyers,        who pointed out that while a full legal transfer to an insurer        of the scheme's obligation to its members would be effective,        holding an insurance contract as an investment asset is an        entirely different matter. Unlike a bond or other securities, an        insurance contract is only a conditional obligation to pay,        subject to their being no defenses available to the insurer.    -   On this basis, the trustees also decided against pursuing other        insurance risk transfer products such as deferred or partial        buy-outs and a product called pension risk insurance, which is        designed to reduce a scheme's deficit and absorb the deficit        volatility for a fixed period of time—ultimately all of these        products would be categorized as an investment in a conditional        obligation to pay (i.e. an insurance contract) rather than a        conventional financial asset like a bond.    -   Another solution which the trustees looked at was a scheme        transfer. This would have involved the transfer of the schemes        liabilities to an independently managed collector scheme, which        in breaking the link to ABC as the employee sponsor would have        removed the troubling deficit/surplus volatility.    -   While superficially attractive, the trustees were quickly put        off this idea after discussion with the UK Pensions Regulator,        which brackets such arrangements under what it terms “scheme        abandonment”. This is strongly discouraged, on the basis of the        Regulator's view that anything which breaks the link with the        sponsoring employer is highly undesirable.

ABC's options employing the inventor's longevity financial instrumentsand methodology

However, innovative capital markets based solutions in accordance withaspects of the present invention provide the following options to thetrustees and the corporate sponsor:

Longevity Indexed Solution

-   -   An embodiment of the present invention would offer ABC an        AAA/Aaa rated longevity indexed security or derivative product        issued from a Jersey cell or Master & Silo company. This product        would make payments according to the actual longevity experience        of a defined population and would most likely be based on the        CMI index, or the newly announced Lifemetrics index, which both        use general population data to generate their indices.    -   To achieve this, the trustees would liquidate existing assets of        the pensions scheme's sufficient to purchase the longevity        indexed securities or derivatives, which would in turn pay to        the scheme an income based on the actual performance of the        chosen index reflecting the actual longevity of the reference        population.    -   The indexed securities or derivatives would provide a hedge        against overall improvements in longevity, so that if people are        generally living longer, the payments would extend accordingly.        The payments on the indexed products would not however track the        specific longevity experience of the ABC scheme's members and        would therefore still leave some potential exposure to longevity        within the scheme—usually referred to as basis risk    -   The indexed securities or derivatives, could be issued for a        specified term, or to the expiry of payments under the index.    -   The indexed securities or derivatives could be issued on a stand        alone basis, or they could also have embedded within their terms        the facility to additionally hedge the specific experience of        the ABC's scheme with respect to inflation, early retirement,        spouse and dependant's pensions entitlements, election to take        lump sums on retirement, transfers out of the scheme, etc.    -   The indexed securities or derivatives could therefore immunize        most of the portfolio risks inherent in ABC's pension's schemes,        but would leave a residual exposure to longevity. The fact that        this solution is based on a generic longevity index, as opposed        to the specific experience of the ABC scheme's would make it a        less costly solution for ABC.    -   The important distinction therefore from any other non insurance        product currently available to pension schemes, is that an        offering in accordance with a preferred embodiment of the        invention is an indexed longevity investment product, which in        addition can encompass the hedging of other scheme specific        exposures, with payments of those elements linked to the actual        experience of the ABC schemes.

Partial Scheme Defeasance

-   -   Alternatively, a further embodiment of the present invention        would enable ABC to opt for a partial scheme defeasance as a way        of solving exposure to its pension schemes. This product, which        can also be offered in the form of AAA/Aaa rated securities or        derivatives can be offered in many permutations, but for the        purpose of this illustration is considered as a 10 year        solution.    -   The pension schemes would liquidate sufficient of their existing        assets to enable the purchase of partial scheme defeasance        securities or derivative contracts with a 10 year maturity.        Income from the securities or derivatives would cover        substantially all of the payment obligations of the pension        schemes for the full period of the investment.    -   At the end of the 10 year life of the securities or derivatives,        there would be a final payment equal to any outstanding deficit        in the schemes, thus ensuring that the scheme's will not only        receive the exact income to meet their obligations to members        for the 10 year period, but that they will be assured of        removing any deficits over the period as required by the        Pensions Regulator    -   The partial scheme defeasance product will benefit ABC and the        pension's trustees by covering substantially all payment        obligations for the life of the investment, eliminating deficits        within 10 years and removing any deficit volatility from ABC's        balance sheet for the life of the investment. However, at the        end of the investment, the full future exposure of the liability        will once again reside with ABC, albeit from a starting point of        a nil deficit. The unknown is what will be the cost in 10 years        of the further hedging which will be required to immunize the        exposures going forward.    -   Partial scheme defeasance as the name suggests is not a total        management solution, but provides a cost effective method of        immunizing deficit volatility and dealing with pensions        deficits.    -   The important distinction therefore from any other non insurance        product currently available to pension schemes, is that an        offering in accordance with a preferred embodiment of the        invention is a bespoke partial scheme defeasance product,        reflecting the longevity experience of the ABC schemes.

Deficit Financing

-   -   While deficit financing is an integral part of the Partial        Scheme Defeasance product, if required an embodiment of the        present invention would also enable deficit financing to be        incorporated into any of the securities or derivative        alternatives outlined in this case study as potential solutions        for the ABC schemes. This means that ABC would be able to fund        its deficit over a period of its choice (up to 50 years), rather        than within the 10 year requirement of the Pensions Regulator.    -   This would be achieved by the issuing cell company or silo        buying a long term debt obligation from ABC equal to the total        size of the scheme's deficits as one of its own investment        assets (suitably credit enhanced and diversified through the use        of credit derivatives). ABC will deploy the proceeds in an        extraordinary deficit filling contribution to its schemes        thereby allowing its schemes to increase the number of AAA/Aaa        rated defeasance securities acquired up to the total liabilities        of the ABC schemes. As a result, both the Pensions Regulator and        the scheme's trustees will be satisfied that the scheme's        deficits have been immediately eliminated.    -   The important distinction therefore from any other non insurance        product currently available to pension schemes, is that an        offering in accordance with a preferred embodiment of the        invention is a bespoke deficit financing product, based on an        exchange of securities issued by ABC for AAA/Aaa rated        investment securities issued to the pensions schemes for the        benefit of its members.

Full Scheme Defeasance—a Total Solution for Pension Scheme RiskTransfer:

-   -   In its most comprehensive form, an embodiment of the present        invention would offer ABC and the pension scheme trustees the        opportunity to invest in AAA/Aaa rated securities or derivatives        issued from a Jersey cell or master & silo company, or a        composite package of securities and derivatives the cash flows        of which would comprehensively reflect the total liability of        the schemes to all of their members.    -   To achieve this, the trustees would liquidate existing assets of        the pensions schemes sufficient to purchase the pension        defeasance securities or derivatives, which would in turn pay to        the schemes an income which in substance defeases the actual        liabilities of the individual schemes.    -   The securities or derivatives would provide a dedicated cash        flow that accurately matches the future obligations of ABC's        pensions schemes to their members, the amounts of which may vary        from current projections due to factors such as actual longevity        experience, inflation, early retirement, spouse and dependant's        pensions entitlements, election to take lump sums on retirement,        transfers out of the scheme, etc. All of these variations in the        actual liabilities of the schemes will however be reflected in        the income received from the securities or derivatives.    -   The important difference between preferred products in        accordance with the invention and any other non-insurance        solutions, is that in addition to tracking inflation and        aggregating the impact of all of the events which affect each of        the individual pension entitlements that make up the scheme,        such as spouse and dependant obligations, election to take lump        sums on retirement, transfers out etc., critically the income        which is paid to the pension scheme will also reflect variations        in longevity of the scheme members.    -   The important distinction therefore from any other non insurance        product currently available to pension schemes, is that an        offering in accordance with a preferred embodiment of the        invention is a bespoke investment product, which is highly        correlated to variations in the actual longevity experience of        the ABC schemes.    -   While this solution will eliminate substantially all of the        variable exposures of the schemes and thereby defease future        obligations to members, any future obligations incurred to        existing or new members of the scheme will not be covered by        this solution. However, the facility exists to add incrementally        to the solution by making further investments in the securities        or derivatives to defease further liabilities as they arise, on        a monthly, quarterly or annual basis.    -   The cost of this solution may be expected to be cheaper than        insurance based solutions, because the products will utilize        debt capital as opposed to equity capital, which as a result of        being issued in risk specific tiers, is both cheaper and more        readily sourced. The result is a more scaleable and cost        effective risk transfer solution.

Impact for ABC of Solutions Based on Inventor's Methodology

-   -   Risk Transfer—Indexed longevity securities or derivatives can        eliminate most of the scheme's exposure to longevity risk, as        well as eliminating other variable portfolio exposures such as        inflation and member specific obligations. This solution will        provide significant risk transfer, but leave some exposure to        longevity basis risk; Partial defeasance securities or        derivatives can immunize the scheme's exposures to both        longevity and other variable portfolio exposures for a fixed        period, at the end of which any scheme deficits will have been        eliminated. However at the end of the investment, ABC would be        exposed to all of the future exposures of the schemes; Total        scheme defeasance securities or derivatives can provide total        transfer of all of the risks inherent in its pension schemes to        the investors in the capital notes which support the structure.    -   Deficit Volatility—The indexed product, if elimination of        non-longevity member specific exposures is included, will        achieve a very substantial risk reduction, but not the        elimination of deficit volatility; The partial defeasance        product will eliminate deficit volatility for a defined period;        The total defeasance solution will achieve the total elimination        of deficit volatility.    -   Trustees—Having purchased any of these solutions the trustees        would remain legally responsible for the schemes (unlike an        insurance buyout, which is a full legal transfer of the        obligation), but they would have the comfort of knowing that the        schemes future obligations to pensioners would be either        partially, or fully covered by a AAA/Aaa covenant, thereby        reducing, or in the case of full defeasance, removing the need        for reliance on ABC's covenant, which is currently        non-investment grade. Depending upon which solution is chosen        the stand alone exposure to ABC's covenant would have been        eliminated or greatly reduced.    -   Accounting—FRS 17 & IAS 19 calculate a scheme's liabilities on        the basis of discounting at the current yield on AA/Aa corporate        bonds and require that where a deficit exists between the assets        and liabilities of a scheme, that the deficit is shown on the        sponsor's balance sheet. The value of AAA/Aaa investment        securities or derivatives would be directly offset and as a        result, in the case of a full scheme defeasance, the value of        the scheme's assets will always match, or exceed the value of        the scheme's liabilities and the problem of the volatile deficit        will be permanently removed. Similarly the partial defeasance        will provide a matching asset to offset the liabilities for the        period of the investment and eliminate volatility. The indexed        solution will substantially reduce, but not eliminate        volatility.    -   Pension Regulator—From the Pension Regulator's perspective, each        of these solutions involving the purchase of AAA/Aaa securities        or derivatives, will meet with approval by raising the certainty        of ABC's pensions liabilities being fulfilled. In addition, if        the solution includes deficit financing, the Regulator's        concerns will also have been met in this regard. The Pensions        Regulator could therefore be expected to lift its restriction on        dividend payments and other distributions to ABC's shareholders.    -   PPF—Similarly, the UK Pensions Protection Fund, which        underwrites the risk of failure by UK corporate pension schemes,        for which it charges a risk based annual levy, can be expected        to respond positively to each of these solutions. Since the long        dated indexed solution or better still the total defeasance        solution would give a greater certainty of the long term        performance of ABC's schemes, over the shorter term solution        provided by partial defeasance, it is likely that they will        result in a greater reduction in the PPF's annual levy, 80% of        which is based on the inherent risks of the specific scheme.        Additionally, where a scheme adopts an AAA/Aaa investment        solution to defease its liabilities, it is expected that,        subject to the language of the documentation, the PPF will give        exemption to the Pensions Act Section 75 provision in the event        of a subsequent failure of ABC, thus protecting scheme members        from a reduction in their pension entitlements.    -   Debt Rating—The impact on ABC's debt rating of any of these        solutions is likely to be neutral, since to the extent that the        solution incorporates deficit financing, it will in effect        replace a long term obligation to the pension scheme with a long        term obligation to the Cell Company or Silo. However, to the        extent that the term of the deficit financing exceeds the        maximum period of 10 years in which the Pensions Regulator        requires the deficit to be eliminated, it may have a positive        impact on the ratings due to the lesser call on ABC's cash flow.    -   Market Timing—One of the most difficult decisions in dealing        with pensions liabilities is timing when to extinguish the risk        exposure. Having taken the decision to utilize a solution in        accordance with aspects of the present invention, one of the        many benefits is the flexibility of the product, which offers        ABC as the sponsor the option (but not the obligation) to invest        in any or all of the tiers of capital which support the AAA/Aaa        ratings of the investment securities or derivatives. By        investing in the capital of the defeasance securities or        derivatives, ABC can continue to participate in the risks and        rewards of managing the pensions risk, which is being        transferred to the capital markets, without further balance        sheet exposure to the volatile pensions deficit—the balance        sheet exposure would now be limited to the size of its        investment in the capital notes. By participating through        ownership of tradable capital securities, which can be sold to        reduce or eliminate exposure at any time, ABC can more        effectively manage the process of extinguishing its pension's        exposure according to its assessment of market conditions. This        is very helpful to a sponsor who is uncertain as to the best        timing of closing out its pension's exposure. The pension        trustees, in the meantime will have the security of holding        AAA/Aaa rated securities or derivatives to meet the liabilities        of the schemes to their members.    -   Future Additions—While the preferred solution only provides        defeasance up to the date of execution—i.e. it is not forward        looking in terms of the accrual of future obligations to ABC's        employees—the solution is flexible in that ABC can subscribe for        additional tranches on a monthly, quarterly or annual basis, so        that future pension liabilities are defeased as they accrue.    -   Sponsor/Trustee Dynamics—There is an inevitable tension between        pension trustees and the corporate sponsor. Trustees want to see        minimum risk and no deficit in the scheme, while sponsors will        typically seek to minimize costs and contributions—especially        for a closed scheme where there may be no way of redeeming a        surplus. The proposed solution in accordance with embodiments of        the invention can uniquely satisfy the demands of both the        trustees and the sponsor, because it provides the flexibility to        fully defease the pension scheme, while leaving the economics        open for the sponsor, if it chooses to participate in the        capital structure.    -   Flexibility—The flexibility of offerings that can be made in        accordance with embodiments of the invention gives ABC the        option to choose different solutions for its different schemes.        ABC might for example opt for full defeasance for those of its        schemes which are in deficit (including deficit financing) to        remove the deficit and its associated volatility. For the        schemes in surplus, it might choose the partial defeasance        solution for a 5 year period, protecting against a swing from        surplus to deficit and thereby neutralizing the volatility for 5        years, with the option to review the position of the surplus        again at a future date. Alternatively, ABC could elect to buy a        full defeasance solution, but for less than the total        liabilities of the specific scheme. In this case it could choose        to reference payments to a defined percentage of the obligations        due to all of scheme members, or just to a nominated cohort of        members of the scheme. The permutation of options available        under a program in accordance with aspects of the invention is        substantial, providing that the exposures are capable of being        hedged, or managed under the criteria agreed with the rating        agencies for the preservation of the AAA/Aaa ratings through the        provision of capital.    -   Covenant—While the preferred solution in accordance with        embodiments of the invention provides the ABC schemes with a        AAA/Aaa rated credit covenant (equal to the strongest sovereign        credit ratings), the facility exists within the issuance program        to add the overlay of an additional independent AAA/Aaa        guarantee from a monoline insurance company or similar entity,        to provide further integrity to the solution.    -   Portfolio Diversification—An important consideration for the        trustees in agreeing to accept a solution in accordance with        embodiments of the invention is that the corporate structure        from which the securities or derivatives are issued is        transparent for the purposes of the requirements for portfolio        diversification required for pension schemes. For this reason        the structure of a trust and sub-trust, or a cell company, or a        master and silo company structure have been chosen to provide a        combination of segregated portfolio exposure (the assets against        which the securities or derivatives are secured are legally        segregated from assets held to secure obligations to third        parties) and “look through”, meaning that the pension trustees        can look through the securities or derivatives, which they hold        on behalf of their members to the underlying diversified        portfolio of assets against which their investment is secured.

Systems Implications

-   -   The proposed solution in accordance with embodiments of the        invention in all of its manifestations relies upon the unique        capability of the systems platform to map the risk inherent in        pension liabilities to the ratings criteria agreed with major        rating agencies for the purpose of securitization and risk        transfer. The systems platform is a vital tool for defining,        monitoring and reporting the relevant risks and for determining        the appropriate levels of capital needed to maintain the debt        ratings for both the senior AAA/Aaa and lower rated junior and        subordinated capital tranches.    -   In addition to its pivotal role in enabling the securitization        of pension liabilities, a further aspect of embodiments of the        invention is that the systems platform can be used by pension        trustees and other managers in the daily management of pensions        risk. Unlike other pensions systems solutions, the systems which        are an embodiment of the invention, provide a comprehensive risk        map of member specific pension liabilities and can uniquely link        the liabilities to the portfolio of pension assets, so that the        risks in the portfolio can be viewed on a holistic basis and at        a granular level.    -   Due to this unique level of functionality, an embodiment of the        invention would allow ABC to adopt the systems platform for its        own internal pension's scheme management purposes. Among the        benefits of doing so would be; the ability to manage its schemes        on a daily marked to market basis for both assets and        liabilities (unlike the present arrangement of revaluing the        liabilities on a triennial basis); the availability of a        transparent reporting system would be of value to ABC's        accountants; it would also provide valuable analysis to equity        analysts and investors, who could be provided with the        information needed to interpret the risks inherent in ABC's        pensions schemes, thereby greatly enhancing their overall        understanding of ABC's business; a further embodiment of the        invention is that it would facilitate transparent reporting of        the composite risks of the ABC schemes to the PPF, which could        be expected to reduce accordingly the risk based aspect of its        annual levy.

Means of Delivering the Solution

Having described the potential solutions which could be provided to ABCthrough the application of various embodiments of the invention, therefollows a brief description of the process by which the varioussecurities and derivatives are issued and managed:

-   -   An embodiment of the invention could involve setting up a Jersey        master company beneath which would sit individual silos, which        would be legally ring fenced from each others obligations    -   The master company would seek and obtain long term debt ratings        for a global multi-currency program of pension defeasance        solutions. The ratings would cover senior and subordinated debt        and capital notes    -   Having been requested by ABC to price a defeasance solution for        one of its pension schemes, all of the scheme member data would        be entered onto the systems platform, which represents a further        embodiment of the invention so as to create a ‘risk map’ of the        scheme. The risk map would be used to derive the amount of        capital required to support an issuance of AAA/Aaa pension        securities, the payments of which would reflect the future        obligations of the scheme to its members. On the basis of this        analysis, ABC would be provided with a price for the pension        defeasance certificates    -   Assuming that the price is acceptable to ABC and its pension        trustees, Silo ‘A’ would issue pension defeasance certificates        in sufficient amount to defease the liability of the scheme. The        monthly payments on the certificates would be the amounts        calculated as sufficient to enable the trustees to meet the        monthly obligations of the scheme to its members, including        one-off payments such as lump sums payable on retirement. The        amounts due would be recalculated on a periodic basis to ensure        that the trustees always have sufficient funds to meet their        obligations—if required, a further embodiment of the invention        would allow the trustees to draw and repay from a liquidity        facility to ensure that they always have funds available to meet        the needs of the scheme. The maturity of the certificates will        be determined by a legal final date, which will be a date after        the scheme has met its final obligations to scheme members.    -   The terms having been agreed, the pension scheme will subscribe        for the pension defeasance certificates, either by exchanging        existing assets of the scheme, or by liquidating existing assets        and subscribing the proceeds to Silo ‘A’ in exchange for pension        defeasance certificates.    -   Having purchased the certificates, which will be secured against        Silo ‘A’s portfolio of investment assets, the scheme will be        required to provide regular updates of member data to enable the        systems platform to monitor the risks profile of the liabilities        which Silo ‘A’ has assumed in issuing the certificates and to        generate the daily rating agency reports required to maintain        Silo ‘A’s debt ratings    -   At the same time as issuing the pension defeasance certificates,        Silo ‘A’ will need to issue sufficient capital notes to satisfy        the rating agencies that it will always have sufficient        resources to meet its obligations, which as a first priority are        to the holders of the defeasance certificates.    -   To enable Silo ‘A’ to meet its obligations, it will invest the        subscription proceeds from the sale of defeasance certificates        in a portfolio of assets, diversified by reference to geography,        industry, issuer and rating, for the purpose of which it will        run its proprietary capital model with updated market data on a        daily basis. The investment parameters under which Silo ‘A’ will        operate will permit investments in short term instruments such        as cash, bank deposits and commercial paper, while at the longer        end of the spectrum, Silo ‘A’ will be permitted to invest in all        forms of fixed income securities together with public and        private equity and alternative asset classes. Every type of        asset and every permutation of asset portfolio will be assigned        a specific capital charge to reflect the risk associated with        the investment.    -   In addition to monitoring and maintaining capital against its        investment portfolio, Silo ‘A’ will be required to monitor its        sensitivities to market risks, such as interest, currency and        inflation and will be required to hedge its exposures to remain        within prescribed tolerances.    -   Finally, Silo ‘A’ will be required to monitor its exposure to        longevity, comparing the actual experience of the reference        population with its own projections of longevity and where        adverse divergence occurs, to provide capital against the        exposure.    -   All of these key portfolio tests will be run daily on the        systems platform to ensure capital compliance and to produce        reports for the rating agencies.

It will be appreciated that the case study and other embodimentsdiscussed above are exemplary only and are not to be taken as limitingthe scope of the invention.

Brief Summary of Conventional Methodologies

The following is the UK Pension Regulator's list of known methodsavailable to manage pension scheme risk, published December, 2006 andavailable on their website.

Buy Out of all or Some Scheme Liabilities with a Regulated Insurer

-   -   Buying out liabilities with a regulated insurance company may        appear to be an expensive immediate exit cost relative to the        cost to the employer of running the scheme on. In practice, this        means the employer is implicitly providing capital from its        business to cover the risks that an insurer has to provide        explicitly. This depends on the appropriate technical provisions        for a scheme linked to the financial strength of the employer,        and the benefit and membership profile of the scheme.

Deferred Buyout of Liabilities with a Regulated Insurer

-   -   Some insurance companies are offering to take on schemes'        liabilities in a phased approach. The aim is that benefits are        insured gradually over time allowing the cost to be spread and        the scheme risks to be managed towards buyout. Some market        entrants are using this to target small to medium sized        companies and schemes that may not have the available capital        for a full buyout.

Longevity Risk Products or Securities

-   -   This covers a range of products or potential products. A        discussion paper on these was presented to the Faculty of        Actuaries in January 2006 (see ‘Living with Mortality: Longevity        Bonds and other Mortality Linked Securities’, D Blake, A J G        Cairns and K Dowd). Existing and past products include over the        counter mortality swaps, mortality bonds limiting catastrophe        risk over the short to medium term issued by a reinsurer to        cover its own life insurance risk, and a longevity bond        announced in November 2004 (subsequently withdrawn).

Primary Layer or Excess of Loss Insurance of Pension Risks Over StatedPeriods

-   -   We are aware of proposals by some companies to insure certain        risk experience within predetermined bands over a stated period        which may be the funding recovery period. For example this may        be to underwrite mortality and investment experience up to a        stated level over the recovery period.

Interest Rate and Inflation Derivatives

-   -   These are primarily over the counter swaps or pooled investment        arrangements provided by investment banks and asset managers.        The intention of these is to improve the match of the scheme        assets to the measurement of the liabilities.

Equity Derivatives

-   -   These usually involve combinations of share options and futures        in order to limit exposure to falls in equity markets. The cost        of these is usually also to limit the potential for equity        gains. These may be used in combination with bond options or        futures to effect a change in the equity/bond mix of the scheme        assets.

Protection Against Employer Default

-   -   Examples of third party insurances include letters of credit and        credit default swaps. A letter of credit provides an amount to        the scheme in the event of employer default as defined in the        agreement. A credit default swap, generally, operates as for a        letter of credit but is a tradable market instrument.

Case Study of Conventional Options Available to Pension Scheme Trustees

The following is a case study analysis of WH Smith pensions defeasance,as published by the UK Pensions Regulator.

In 2005, faced with a £100m deficit, WH Smith took a radical step to tryto deal with the problem.

Why Did they Decide to Change their Investment Strategy?The trustees took advice from an investment bank, which analyzed thefund in relation to risk. This showed that the fund was exposed to threetypes of risk: equity risk, interest rate risk and inflation risk.The trustees decided that they wanted some, but not a lot, of equityexposure but no interest rate or inflation risks. They were alsoconcerned that equity was an imperfect match for their pension fundliabilities.Their pension payments were inflation linked. The trustees wanted tochange their investment strategy so that it matched these liabilities.What Did they do?They invested 94% of the fund in swaps (inflation and indexed linked).The remaining 6% was invested in options, which allowed the scheme someequity exposure.The WH Smith trustees took a lot of advice before deciding upon thisliability driven investment strategy. They considered 30 differentmodels provided by banks and fund managers before making a decision.

What was the Result?

This strategy proved at least partially successful for WH Smith. Theiradvisers said that if the trustees had kept their original strategy, thedeficit would have increased to £150m because of the unprecedented fallin bond yields in early 2006.The timing of strategies like this is key. WH Smith's strategy was putin place in October 2005 before the further fall in bond yields.

A Final Note.

Despite this strategy, in January 2007 WH Smith announced that it neededto close the scheme even to existing members. The company stated that‘the long term costs of running a final salary scheme continue to behigh and difficult to predict, mainly due to low investment returns andmembers living longer.’

Inventor's Capital Markets-Based Solution

The lesson which trustees may draw is that liabilities can be veryunpredictable, even in the short term.

The basis of one aspect of the invention is that it provides a structurefor a defeasance product which creates minimal basis risk for the issuerand the investor, assuming that the investor is seeking to defeaseexposure to actual pension liabilities rather than exploit relativevalue.

For ease of understanding, the comments below refer to a securitisedembodiment of a product in accordance with this aspect, but they canalso apply to the derivative form.

Method of Initially Setting and Later Re-Setting Payment ScheduleAmounts

An embodiment of a method according to the present invention ofoperating a financial instrument associated with the defeasance of apension scheme will now be described.

The design rationale of the method of operating the defeasance productto project cash flows and also of calculating the indexed cash flowsthat make up the adjusted cash flow to be paid at re-set points to theinvestor holding a financial instrument according to the inventionconcerns two aspects, life expectancy (i.e. mortality experience) andpension cash flow (taking into account non-mortality experience) willnow be set out.

In this embodiment, the life expectancy construct of the defeasanceproduct has been based on the understanding that its cash flows may bedetermined by reference to the actual average or weighted averagemortality rate of a defined population or sub-population (i.e.“segment”) of scheme members, but may not be determined by reference tothe deaths of individual scheme members. In other embodiments, referenceto the deaths of individual scheme members may be made.

The pension cash flow construct of the defeasance product has been basedon the understanding that cash flows attributable to each scheme membermay be varied due to non-mortality events such as pension, work orlifestyle choices of individual scheme members (e.g. election for a taxfree lump sum on retirement, pay increase, marriage etc) but may not bevaried due to the death of individual scheme members (i.e. mortalityexperience).

The defeasance product requires the calculation of two sets of cashflows, projected cash flows and indexed cash flows, together with a ratere-set on a regular basis.

The working assumption is that, depending on the scheme, rates will bere-set on a monthly, quarterly or annual basis (each a “rate re-setperiod”). The adjusted payment amount to be paid to the investor in thatperiod is thus calculated in accordance with the rate re-set method.

Projected cash flows will be calculated prior to each issue of afinancial instrument, such as defeasance securities, in accordance withthe present invention. Based on personal and statistical data availableat time of issue, the capital projection model will project future cashflows for the scheme as a whole, all segments of the scheme, and everymember of the scheme. The likelihood that each pension scheme memberwill survive until given times in the future projected by an appropriatestatistical longevity projection model may be taken into account incalculating the projected cash flows prior to the issue of the financialinstrument.

These projected cash flows calculated for the scheme as a whole will bethe issuer's monthly scheduled payment obligations on the securities andwill be documented as such in the relevant pricing supplement for eachissue.

The statistical data used for life expectancy/longevity projections mayalso be based on appropriate actuarial tables as amended for thedemographic and socio-economic characteristics of each scheme, segmentand member. These amendments of the longevity tables for each member ofthe pension scheme, or ‘mortality level adjustments’ will be describedlater.

At each re-set point after the financial instrument has been issued,indexed cash flows will be calculated in relation to the rate re-setperiod just completed. Based on updated personal and statistical datarelated to the pension scheme's actual experience, the model willre-calculate cash flows for the rate re-set period just completed forthe scheme as a whole, all segments of the scheme, and every member ofthe scheme.

The indexed cash flows for any rate re-set period will comprise theaggregate value of indexed cash flows for all segments of the scheme.The aggregated indexed cash flow represents the adjusted cash flowamount paid to the investor in relation to that re-set point.

The statistical data used for life expectancy outcomes during thatmonth, quarter or year is based on the actual average mortality rate foreach segment of the scheme.

A cash flow entitlement is calculated for every original member of thescheme regardless of whether they are alive or dead.

All other (i.e. non-mortality related) personal and statisticalinformation used in the calculation of indexed cash flows will be basedon actual (rather than projected) data. For example, the model will useactual RPI growth for indexed pensions, actual tax free lump sums and“transfers out” during the rate re-set period (i.e. members portingtheir pension entitlements to a different scheme).

Rate re-sets will take place each month, quarter or year and willinvolve increasing or decreasing the issuer's monthly, quarterly orannual scheduled payment obligation on the securities by reference tothe net difference between the Projected and Indexed cash flows for thatmonth, quarter or year.

Where, during the rate set period under review, only one scheme memberhas died, the element of the rate re-set calculation attributable tomortality data will be deferred until the rate re-set period duringwhich the next scheme member dies. The deferred rate re-set in respectof mortality data will be calculated for the composite period from thefirst day of the original rate re-set period to and including the lastday of the deferred rate re-set period. All scheme members dying duringthis composite period will be deemed to have died part way through thecomposite period using a time based weighted average.

This “single death” procedure will only apply on a scheme wide basis; itwill not apply to a single death in a segment of the scheme if at leastone other scheme member from one of the other segments of the scheme hasdied during the rate re-set period under review.

Run Off: if the issuer has not previously redeemed the defeasancesecurities (perhaps by exercising its “clean-up” call option) and fewerthan 11 members of the scheme are still alive, the issuer will berequired to redeem the securities by paying investors an amount equal tothe cost of buying annuities for all of the remaining members.

An optional feature of the product is that if rate re-sets occur on aless regular basis than payments on the securities (e.g. quarterly orannually rather than monthly) the issuer may provide deposit andliquidity facilities to an investor to help “smooth” differences betweencash flows received on the defeasance securities and payments due toscheme members.

In a preferred embodiment of this aspect of the invention, “segments”will be created based on the status of each member (deferred, active,pensioner) and size of pension entitlements in each rate re-set period.Table 1 shows defined segments for members of a large scheme sorted byreference to £200 annual pension entitlement bands. This would produceat least 1,000 segments for a scheme of say 50,000 members and annualpension entitlements ranging from £0 to £200,000. Although thiscorresponds to an average of 50 members per segment, the average willcover a wide range of segment sizes, some of which may be severalhundred strong others of which may be empty or just have a handful ofmembers.

TABLE 1 Segment   1 > £0 < £200   2 > £200 < £400   3 > £400 < £600  4 > £600 < £800   5 > £800 < £1,000   6 > £1,000 < £1,200   7 > £1,200< £1,400   8 > £1,400 < £1,600   9 > £1,600 < £1,800  10 > £1,800 <£2,000  491 > £98,000 < £98,200  492 > £98,200 < £98,400  493 > £98,400< £98,600  494 > £98,600 < £98,800  495 > £98,800 < £99,000  496 >£99,000 < £99,200  497 > £99,200 < £99,400  498 > £99,400 < £99,600 499 > £99,600 < £99,800  500 > £99,800 < £100,000  991 > £198,000 <£198,200  992 > £198,200 < £198,400  993 > £198,400 < £198,600  994 >£198,600 < £198,800  995 > £198,800 < £199,000  996 > £199,000 <£199,200  997 > £199,200 < £199,400  998 > £199,400 < £199,600  999 >£199,600 < £199,800 1000 > £199,800 < £200,000The worked example below exemplifies the contrast between the respectiveobligations of the pension trustees to its members and the issuer to itsinvestors consequent upon the death of scheme members. For example, withmore than 250 deaths by the year 10 rate re-set, the pension trustees'future obligations to their scheme members are now based on the pensionentitlements of fewer than 750 surviving members. By contrast, theissuer will always calculate payments on its defeasance securities byreference to the original population of 1,000 members. Even though morethan a quarter of members have died by the end of Year 10, the issuerwill still calculate and pay cash flows on its defeasance securities inrespect of all 1,000 original members, whether alive or dead based oncumulative projected and average actual mortality data for the 10 yearperiod.

Example Application of Payment Schedule Setting and Re-Setting Method

A hypothetical scheme overview will now be described, with reference toFIGS. 6-16.

In this example, there are 1000 members labelled 001 to 1000. Some arealready retired (retirement year 0), while others retire up to 13 yearsinto the future. The scheme members have differing initial (annual)pension entitlements ranging from circa £1K to £31K, and differentexpected mortality rates (based on age, gender etc.). There is anindexation scheme—4% pre-retirement, and RPI after retirement.

In accordance with this embodiment of the invention, people will beassigned to a segment in any given year based on their nominal pensioncash flow in that particular year. In this example, the segments aredefined in terms of £1,000 intervals. Thus, for example, segment 1 inyear 5 would consist of those individuals whose annual pension in thatyear ranges from £1,000 to £2,000.

As of time zero, when the financial instrument is to be issued, it ispossible to project each member's nominal cash flow, as shown withreference to FIG. 7. The projections are based on an RPI of 3%. Pensionsgrow due to indexation before retirement of 4% and RPI afterwards, butonly get paid from retirement date.

The calculation of the expected cash flows is performed as follows, withreference to FIG. 8. As described above, it is possible to predict eachmember's expected cash flow, taking account of their projectedprobability of death. The probability of death/survival of each memberas at time zero, when the financial instrument associated with thepension scheme is to be issued, is projected using the longevityprojection model in accordance with aspects of the present invention, tobe described below. It can be seen in FIG. 8 that there is provided theprobability that a member will die before a given date, as at time zero.Multiplying the nominal cash flow by the probability that the member isstill alive, provides expected cash flows. This expected cash flow formsthe payment schedule for the bond that is issued. These expected cashflows will be the issuer's scheduled annual payment obligations on itsdefeasance financial instruments such as securities. This is how thescheduled payment amounts of the financial instrument match the expectedcash flow obligations of the pension scheme to its members.

There will now be described how indexed cash flows and rate re-sets canbe calculated to take account of actual experience for all factors otherthan mortality, and a segmented average of actual experience formortality.

After the financial instrument is issued, adjusted payment amounts arecalculated at regular re-set points. As an example, the calculation ofthe year 10 rate re-set will be discussed, with reference to FIG. 9. Itcan be seen that in year 10, two things have turned out differently towhat was expected. Firstly, RPI turns out to have been 4% and not 3%,and secondly member 002 commutes 20% of his pension. This leads to a 20%drop in his pension entitlement compared to what it would have been, buta spike in his cash flow due to the lump sum payout. It will be notedthat all 1,000 scheme members are put into year 10 segments based ontheir pension entitlement, regardless of whether they are still alive ornot. This segmentation makes it possible to place all 1,000 into asegment. Thus, as shown in FIG. 10, all 198 deferred members are insegment 0, while the retired members are in the segment according totheir pension entitlement in year 10. As shown in FIGS. 11 and 12, it ispossible to allocate the revised nominal cash flows to each segment, andto calculate the average survival rate for each segment. Whetherrevising cash flow projections or mortality projections, the revisionsare always based on the original 1,000 members, regardless of whetherthey are alive or dead.

There is now all the information needed to calculate the Year 10aggregated indexed (i.e. adjusted) cash flow. With reference to FIG. 13,the revised nominal cash flows are multiplied by the average survivalrates, and the segments are added up to provide the Year 10 aggregatedindexed (i.e. adjusted) cash flow. As shown in FIG. 14, it is thenpossible to calculate the Year 10 rate re-set, being the differencebetween the projected cash flows and the aggregated indexed (i.e.adjusted) cash flows.

To understand the basis risk, it is possible to work out what cash flowsthe trustees actually need to pay to their members. With reference toFIG. 15, it is possible to look at which individuals actually died. Inthe actual mortality experience table, an entry of 100% means that theindividual died; otherwise the entry is 0%. It is possible to calculatethe cash flows the trustees actually need to pay out to each member, asshown in the cash flow required table. The cash flows are calculated asthe member's nominal cash flow (but on an actual basis as describedabove with reference to FIG. 9) if he/she survives, and are set at zerois he/she does not. For year 10, the cash flow required in this exampleis £11,889K, and the right hand side of FIG. 15 shows the cash flowssplit into segments.

The difference between what the issuer pays investors on the defeasancesecurities and what the trustees need to pay the scheme members, is thebasis risk. With reference to FIG. 16, it can be seen that segment 2shows no basis risk because it has no mortality experience. In general,no basis risk arises whenever either no members or all members of asegment have died. It can be seen that the difference between theindexed cash flows and the pension cash flows in this example is£11,000. Thus, investors in the longevity instrument will receive£11,000 more than the trustees actually need to pay the scheme members.This represents a 9.3 basis points basis risk exposure on the 10m yearrate re-set.

Method of Projecting Longevity and Quantifying Longevity Risk

There will now be discussed in some detail the methodology forprojecting the longevity of a pension scheme membership and also sizingand capitalising longevity risk, in accordance with aspects of theinvention.

The methodology according to aspects of the invention described hereinfor sizing and capitalising longevity risk can be used to quantify andprice the longevity risk associated with a pension scheme due to theuncertainty associated with the future mortality experience of thepension scheme's members, and thus also the longevity risk associatedwith a financial instrument according to the present invention whichtransfers the longevity risk of a pension scheme onto the capitalmarkets. This can assist investors in understanding the longevityexposure of the financial instruments of the present invention.

The methodology of the present invention can also be applied todetermine an amount of risk capital to be held to support a financialinstrument according to the present invention so that it achieves andmaintains a rating according to criteria agreed with a ratings agency.The risk capital can be held in the form of subordinated tranches ofdebt and equity, issued in the form of, for example, capital notes andequity notes.

The methodology of aspects of the present invention can also be appliedgenerally to quantify the longevity risk exposure of any asset or aliability having cash flows of sums of accounts receivable and accountspayable which are dependent to some extent on the actual futuremortality experience or exposure of a group of creditors or debtors.

Modelling and Adjusting Mortality Tables for Longevity Trends

Longevity ‘trend’ risk is the risk that the trend in mortality rates isdifferent to that expected, i.e. people live longer than projected. Therate of mortality improvement has been increasing over time, driven byincremental improvements in medical advancements, rising standards ofliving and generally healthier lifestyles. Also, certain age groups haveseen higher rates of mortality improvement than others. This phenomenon,known as the “cohort effect”, has resulted in the actuarial professiondeveloping more robust statistical techniques to predict futurelongevity.

The P-spline model is a statistical technique that has gained wideacceptance to date both within the industry and across academia andpreferred embodiments of aspects of the present invention utilise theP-spline model to forecast longevity. However, any suitable statisticallongevity projection technique may be utilised, such as, Cairns, Blakeand Dowd's model and the Lee-Carter model. The following will bediscussed below: the development in the Continuous MortalityInvestigation (CMI) bureau Working Papers of the P-spline model forforecasting longevity, how to use the P-spline model, which data sourcesare appropriate to provide a suitable reference population and keyissues to be aware of regarding the use of the model. The practicalissues surrounding longevity and a step-by-step process for producing amortality table will also be discussed. Overall, the P-spline modelprojections are more conservative (i.e. project greater improvements inlongevity) and are generally accepted to be more accurate thanpreviously published projections by the CMI.

In preferred embodiments of aspects of the present invention, once themortality tables with the future mortality rate projections have beenconstructed according to the P-spline model, the capital requirement tocover longevity risk exposure of a financial instrument according to thepresent invention can be estimated. In preferred embodiments, thecapital requirement is calculated by ensuring sufficient capital is heldso that the liability is covered in the worst case longevity scenario.The worst case scenario is calibrated in differing ways depending onwhether the product being offered is to be rated by either Standard &Poor's and Fitch or Moody's ratings agencies.

If the product being offered is to be rated by Standard & Poor's orFitch to have a certain rating, the worst case scenario is preferablycalibrated to the default probability of an equivalently rated bond.

If the product being offered is to be rated by Moody's to have a certainrating, the worst case scenario is preferably calibrated to the expectedloss of an equivalently rated security.

The drivers of longevity improvements will now be discussed.

Over time we observe that mortality rates decline, and so averagelife-spans increase. This trend is driven by a combination of factorsincluding incremental improvements in health care, rising standards ofliving (for example better insulated housing), changing lifestyles (forexample a decline in smoking rates), and incremental public healthinitiatives (for example stricter regulation of air pollution). All ofthese drivers tend to result in gradual declines in mortality ratherthan step changes. For example, we typically observe that rather thaneliminating broad classes of diseases in one go, new drugs tend to beeffective against narrow classes of illness (e.g. one form of livercancer) or deliver a higher success rate than their predecessor.Consequently the rise in longevity should be viewed as the compoundingeffect of a large number of incremental improvements. This longevityimprovement trend is illustrated in FIG. 17 which shows the annualpercentage decline in mortality rate (q(x)) for the male population aged20-90 based on smoothed data from the Office of National Statistics(ONS). If q(x) is the mortality rate, then FIG. 17 shows the percentagedecline in the mortality rate from one year to the next.

Since the mid 1970s, there has been observed an acceleration in the rateof mortality improvement in the UK. FIG. 18, which breaks out this rateof improvement and shows the annual percentage mortality decline byage-group for males aged 55, 65 and 75. FIG. 18 shows that they havebeen particularly driven by specific groups of people: 55-year olds inthe late 70's and early 80's; 65 year-olds in the 90's and 75-year oldsnow. This reflects a phenomenon known as the ‘cohort effect’ whichobserves that the cohort born between 1925 and 1945 experiencedespecially marked improvements in their longevity. Of course thesubsequent generations that followed this cohort would show relativelylow rates of mortality improvements being measured off the low mortalityrates of the 1925-45 cohort. But the 1925-45 cohort also sawsignificantly greater improvements in mortality (relative to theirpredecessors) than prior cohorts. In preferred embodiments it is,important that the “cohort effect” is accounted for in projectingmortality rates.

The P-spline methodology for projecting improvements in longevity inaccordance with preferred embodiments of aspects of the presentinvention will now be described. The use of a statistical longevityprojection model, such as the P-spline, according to aspects of thepresent invention is to project trends in improvements in mortality in asuitable reference population in order to produce individual mortalitytables to project with greater confidence the mortality of each of themembers of the pension scheme into the future, and to calculate theexpected cash flows of the financial instrument and the capitalrequirement on that basis.

Historically, pension liability valuations were based on mortalitytables produced from actual experience and a flat mortality assumptionwhich does not allow for improvements in mortality. Such an approach wasfound not to be conservative enough and, as a result, these mortalitytables have been extended to allow for mortality improvements linked tothe ‘cohort effect’. These interim adjustments to the tables wereessentially to ‘roll forward’ the trend improvements in longevity seenin previous years. However, these adjustments tend not to be grounded inrigorous statistical theory but are based on expert judgement byactuaries and are subjectively set by choosing a range of projectionbases. These arbitrarily chosen tables have been found to probably benot sufficiently prudent.

In view of this, the industry has recently taken major steps forward inestablishing more rigorous statistical underpinnings to mortalityprojections. The P-spline model is the statistical technique that hasthe widest acceptance among industry experts, academics and the largerand more sophisticated insurance companies. In various aspects of theinvention, the P-spline is the preferred methodology for projectinglongevity. However, further research is continuing on a range of otherstatistical models such as the Lee Carter model, or the Cairns, Blakeand Dowd model, which may also be used to project longevity inconjunction with the present invention.

A spline is a function defined piecewise by polynomials. Splines aregenerally used for interpolation or smoothing of data sets (e.g. toderive a complete yield curve using points on the curve). Furtherdiscussion of splines in general can be found in Eilers P and Marx D.,‘Flexible smoothing with B-splines and penalties’, Statistical Science,Vol. 11, No. 2, p. 89-121, 1996. Further detail on the application ofP-splines specifically to mortality data can be found in Currie I.,Durban M. and Eilers P., ‘Using P-splines to extrapolate two-dimensionalPoisson data’, Proceedings of 18th International Workshop on StatisticalModelling, Leuvan, Belgium, p. 97-102, 2004, and CMI, ‘Projecting futuremortality: Towards a proposal for a stochastic methodology’, Workingpaper 15, Jul. 2005. These documents are incorporated herein byreference.

In general, when fitting polynomials to observed data, the higher thedegree of polynomial that is used, the better the fit. However, it isnot always desirable to use a high degree polynomial as this can oftenlead to ‘over-fitting’ and to poor predictive stability outside of theobservation period. Instead, the P-spline calculates what is known as a‘penal spline’: by applying a penalty to increasing degrees ofpolynomial it trades off parsimony in estimated coefficients foraccuracy of fit. If we choose a small penalty we follow the dataclosely, and the possibility of over fitting is in this case lurking. Onthe other hand, choosing a very large penalty leaves very little roomfor following the data. There is a trade-off between smoothness-of-fitand goodness-of-fit. Any of the common criteria for optimisingsmoothness versus goodness-of-fit can be used, such as the BayesianInformation Criterion (BIC) or the Akaike Information Criterion (AIC).

The other way in which a P-spline differs from a simple spline is thatit can be carried out over two dimensions. In other words, rather thanfitting a curve to a set of observations, the P-spline fits a surface toa two-dimensional array defined by age and year of observation.

One important choice which needs to be made in using a P-spline iswhether to use an age-period or age-cohort spline. The former projectsmortality rates based on historical patterns observed by age group andyear of observation. The latter projects mortality rates based onpatterns observed by age group and by cohort. In the invention, thestrong cohort effect which is apparent in the UK longevity data leadsthe age-cohort model to be preferred. The age-cohort central projectionhas be found to be more conservative than the age-period centralprojection.

The P-spline model reads in data on historical observations for deathsand for the population as a whole, and fits a P-spline to the resultingdeath rates. The model then projects the P-spline forward in time todeliver projected mortality rates into the future. Finally, the modelalso delivers standard errors of the fit, indicating the goodness offit. In aspects of the invention these standard errors are then used toestimate the capital requirements to cover longevity risk.

An example of using the P-spline methodology to project longevity in theUK dataset in accordance with a preferred embodiment of the inventionwill now be described using the age-cohort model only.

The goal is the construction of a longevity mortality table and thisprocess will be described step by step. In this example, the approachtaken to P-spline modelling in CMI Working Paper 20 (which isincorporated herein by reference) is followed. This paper concludes thatthe interim cohort projections show a lower pattern of observedmortality improvements in comparison to the P-spline model. The P-splinemethodology is better able to project forward the actual improvements asmore recent data becomes available. However, it cautions that care isneeded in the choice regarding the dataset selected for a referencepopulation and the parameters and penalties used.

When running the P-spline model, key considerations are the selection ofan appropriate data set and P-spline knot placement. A knot is a pointwhere the polynomials making up the P-spline are joined. The CMIrecommends a minimum of 20 consecutive years of data spanning an agerange of at least 40 years. Additionally, there needs to be sufficientnumber of deaths and exposures for each age in each year. A minimumnumber of 1,000 lives (exp) and 30 deaths in each data cell by year andage is preferred. The knots of the P-spline should be placed to ensurethat no polynomial piece in the fitted splines spans both the data andthe projected region. The best way to ensure this is to place knots atthe leading edge of the data. Knot locations will need to be changed asnew years of data become available and the model is updated.

The first step is to select an appropriate data set of actual mortalityexperience for a reference population for which the P-spline model canbe used to project mortality improvements. The data set should meetthese minimum requirements outlined above. In the UK, for example, thereare two main sources for mortality experience data:

1. Continuous Mortality Investigation (CMI) provides mortality data formale assured lives from 1947 to 2005 covering ages 11 to 100. The CMIstarted collecting female data from 1975 to 2005. However, prior to 1983the data was collected in aggregate age and year bands. Furthermore, thedata is very limited at higher ages (above 70 years). These factors makethe female data unreliable for mortality projection. The CMI data coversthe UK insured population which is generally a more affluent segment ofthe total population. The insured population has lower mortality ratescompared to the UK population and have experienced stronger mortalityimprovements in the past.2. Office for National Statistics (ONS) provides mortality data for thepopulations of England and Wales from 1841 to 2003 for ages 0 to 110 forboth males and females.

In this example, the CMI data has therefore been used for projectingmortality as this better reflects the population underlying theliabilities of an exemplary pension scheme and is more prudent. However,special treatment has to be made for female mortality projections. thefemale CMI data set does not have a large enough population at high agesto be considered. Therefore, to provide mortality rates for females thefemale ONS data set and the male ONS data set were also analysed. Thefemale improvement factors can therefore be calculated by taking thedifference between ONS male improvement factors and ONS femaleimprovement factors to adjust the CMI male improvement factors.

In the CMI data set only ages 20 to 90 are used for projecting mortalitybecause members of a pension scheme will be of working age, makingmortality projections for younger ages irrelevant. Ages above 90 are notconsidered due to small exposures at these ages. Although the ONS dataset is a larger data set both in time spanned and number of livescovered there are some difficulties in applying the data set to data inthe early years. Years prior to 1953 have not been considered due todifficulties with the data especially around World War I and World WarII. Specifically, some approximations and estimates had to be made tothe number of deaths in the periods 1914-1920 and 1939-1949 due to lackof accurate data. That said, there is more than 50 years of data to workwith, which is sufficient for projection purposes.

Next, the step of running the P-spline model to project mortalityimprovements in the selected reference dataset will be discussed.

In this example, CMI's recommended default parameters and calibrationdata set (covering ages 21 to 90 and years 1947 to 2005) have been used(see CMI Working Paper 27, July 2007, incorporated herein by reference).For all of the P-spline fits, cubic splines and a penalty order of twohave been used. The knots have been placed on both corners of theleading edge of data. In practice this means that there are knots at age21 and 90 and on the last year of data. The projections have beenperformed for 100 years into the future, e.g. to 2105 for base yearprojections from 2005. Changing the number of years projected may affectthe fit.

Table 2 presents the parameters used for the age-cohort penalties modelbased on the data set that results in a high goodness of fit and prudentresults.

TABLE 2 CMI assured lives males Calendar year range 1947-2005 Age range21-90 Knot spacing: Age dimension Every 3 years Cohort dimension Every 3years

Several sense checks are run on the model's output; the model outputsthe Bayesian Information Criterion (BIC) which is optimised over thepenalty weights (the lower the number the better). Analysis confirmsthat, when using the CMI data, a BIC of 7,600 has produced reliableresults. This also agrees with results of a study by Cairns et al whoproduced a BIC number of 9,300 on a slightly different data set whenusing a P-spline model. A second check performed is to verify that thestandard error (S.E.) terms are not excessive over the whole period, butmost importantly to monitor the later years in the projection.

Next, to produce the projected mortality table, the projectedimprovements in longevity from the P-spline model are applied to a basemortality table for that reference population. In this example, the basetable is the latest full table published by the CMI, the PNMA00 table.This is defined as the Life Office Pensioners, Males, Normals for theyear 2000. This mortality table is fitted to the combined mortalityexperience of all pension business written by insurers including bothdeferred and immediate pensions. The year-on-year improvements from theP-spline model are then applied from this year going forward.

FIG. 19 shows a comparison of the weighted average P-spline modellongevity projections for males aged 55-90 and the previous CMIpublished projections (the ‘Medium Cohort’ table). Overall, the P-splineprojections are more conservative (i.e. project a greater rate ofimprovement in mortality rates) than the Medium-Cohort projections.

The resulting improvements in mortality projected by the statisticallongevity projection model, such as the P-spline model used in thisexample, can be validated by carrying out a ‘what if’ or back testinganalysis. This can be performed by using statistical longevityprojection model to fit data at a point in past history and assess theadequacy of the best estimate capital requirement of a sample portfolioof pensioners and the worst case capitalisation at a required (AAA/Aaa)confidence level by comparing the projected liability with the actualliability in the full-run off of the portfolio of pensioners usingactual mortality experience for the projected period.

The resulting improvements in mortality projected by the statisticallongevity projection model can also be validated by performing acomparison of those results with the results of a qualitative analysisof the trends in mortality improvements in the reference population.This qualitative analysis may take into account the effect on longevityof factors such as historical longevity trends, uncertainty,socio-economic factors, behavioural factors, gender issues, mortality bycause of death, and medical discovery risk. For example, one can ask thequestion, what would be the improvement in longevity if there were asignificant reduction in obesity or a cure for cancer were suddenlydiscovered, and compare those effects with the projections of thequalitative model. The qualitative analysis thus far conducted on theCMI data for the UK have confirmed the appropriateness of and outputsfrom the quantitative forecasting models of embodiments of aspects ofthe present invention.

Modelling and Adjusting Mortality Tables for Mortality ‘Level’

The methodology of aspects of the present invention of taking intoaccount the mortality level risk associated with the particular pensionscheme membership in the calculation of mortality projections for theindividual pension scheme members, in addition to the trend projectionsdescribed above, will now be described.

Level risk is the risk that a particular pension scheme membership has adifferent level of mortality risk compared with that of the referencepopulation on which the mortality table incorporating the quantitativemortality trend projections is based. In the example given above, thatis the risk that the pension scheme membership has a different level ofmortality risk compared with that of the general UK insured populationas a whole (as evidenced by CMI mortality data) which forms the basisfor the longevity projections.

The approach taken to level risk is granular in that, in embodiments, itinvolves analysis of life expectancy profiles based on fullpostcode/zipcode geographical analysis, and where possible drilling downto residents of individual households. That is, there is no averagingassumption and the impact of mortality level differentials isincorporated at the level of the specific pension cash flows ofindividual members, and the approach is thus granular.

Mortality level adjustments are calculated for every individual in thereference portfolio (by reference to his or her age, sex, lifestyle,pension size and even postcode) and incorporated into each individual'spension cash flows—i.e. at the most granular level possible. These leveladjustments are produced as a result of an analysis of the effects onmortality of the different socio-economic factors and the calculatedadjustment for each member may be incorporated into the mortality tableproduced by the statistical longevity projection model by way of amultiplication factor, an addition, a subtraction, or some otherfunction of varying the mortality rate contained therein.

As for quantum, the aggregate of all level adjustments has been found tohave a small impact on the Net Present Value of a reference portfolio'sliabilities—the impact may vary from one reference portfolio to anotherbut is likely to be less than 5% for the majority of pension schemes, onthe basis of current studies.

The socio-economic characteristics that are taken into account in thelevel risk adjustment may be at least one of the following: age, gender,pension size, socio-economic class, smoking status, geographicallifestyle mapping, zipcode/postcode, seasonality based on date of birth,taxation level, real estate ownership level, family status, maritalstatus, number of dependents and occupational industry.

Longevity Capital Assessment Methodology

The longevity capital assessment methodology of aspects of the presentinvention which is used to estimate the capital requirement to coverlongevity risk will now be described.

As discussed above, this estimation of the capital requirement iscalculated by ensuring sufficient capital is held so that the liabilityis covered in the worst case longevity scenario and that the worst casescenario is calibrated in differing ways depending on whether theproduct being offered is to be rated by either Standard & Poor's andFitch or Moody's ratings agencies. Thus the preferred approach tolongevity capital assessment for Standard & Poor's and Fitch ratedfinancial instruments will be discussed first, followed by the preferredapproach for Moody's rated financial instruments.

Ratings Method for Quantifying Longevity ‘Trend’ Risk

For a Standard & Poor's or a Fitch rated financial instruments, inpreferred embodiments of aspects of the invention, the approach is tohold sufficient capital to ensure that the probability of default (i.e.cumulative probability of default) is lower than that observed forcorporate bonds of the target debt rating. Thus, the capital estimatesof the longevity capital assessment are anchored on a calibration ofStandard & Poor's or Fitch's rated corporate bonds. In the example givenbelow, the estimation of the capital requirement to achieve a ratingfrom Standard & Poor's rating agency is described.

FIG. 20 shows the estimated default probabilities, which are derivedfrom Standard & Poor's data for AAA, AA, A and BBB rated corporate bondsand extrapolated beyond 15 years based on the appropriate ratingtransition matrices. As one would expect, these rise over time. In theinvention, the approach is to ensure sufficient capital is held so thatthe default probability is lower than the relevant bond class at alltime horizons. In this sense the capital calibration according to theinvention is very conservative since at all horizons other than thebinding time horizon our default probability will be lower than that ofan equivalently rated bond.

In order to calculate the capital requirement for any given timehorizon, two different approaches are possible, both are within thescope of aspects of the invention.

The primary and preferred approach is the deterministic approach, whichis based on applying stress tests of the appropriate size to the cashflow projections and observing the resulting impact on liabilityvaluations.

The other approach is the stochastic approach, in which stochasticlongevity shocks are simulated and the portfolio is re-valued for eachone. By observing the tail of the resulting distribution we cancalculate the required capital.

Both of these approaches give the same estimate for economic capital.However, the advantage of the stochastic approach is that it providesgreater flexibility—for example allowing us to estimate the ‘tail’value-at-risk (VaR) as well as straightforward VaR. These two approacheswill now be described in turn in more detail.

Deterministic Longevity Trend Risk Quantification Method

For any given time horizon, the deterministic approach essentiallyinvolves answering the question “How much capital do we need to hold towithstand the worst case shock which arises with a probability of nomore than x %?” where x is our target default probability for the giventime horizon. So, for example at a 5 year time horizon, we know that thedefault probability of a AAA rated bond is 0.10%. Therefore, if we canidentify the longevity shock which arises with this probability then wecan use this to calculate how much capital is needed.

The required capital is then calculated as the difference between the‘Best estimate’ value of the pension liabilities and the shocked valueof the pension liabilities at the relevant confidence interval. FIG. 21illustrates this calculation.

As discussed above, an advantage of the P-spline approach used inpreferred embodiments of aspects of the invention is that as well asproducing a ‘best estimate’ of future mortality rates, it also producesconfidence intervals around that best estimate. As an example, FIG. 22shows these confidence intervals for a 65-year old male. In the bestestimate, the annual probability of death drops from 88 basis points(bps) to 74 bps over the first 5 years. But in the worst case it dropsto 68 bps.

To apply these stressed mortality scenarios to the capital calculation;there needs to be estimated what the impact of such a shock would beover the relevant time horizon. Taking again the example of a 5 yeartime horizon, FIG. 23 shows for a 65-year-old male the two impacts that5 years of shocked mortality experience would have on our liabilityvaluation.

The first impact is via the lower mortality experience during those 5years. The fact that fewer people than expected die in years 1-5 meansthat more pension payments have had to be paid out during those yearsand, other things being equal, more will have to be paid out in futureyears for the people who were expected to die during years 1-5 but whodidn't.

The second impact of the shock is via its effect on the assumptions madeabout future mortality rates. If lower mortality rates are observed overa sustained period of time, the future longevity projections also thenneed to be revised. Therefore the P-spline model must be re-run at year5, taking account of the bad news experienced from years 1-5 as well asthe historical data prior to that. These revised expectations are shownby the ‘revised best estimate’ line in FIG. 23.

The capital required to withstand the worst case shock over a 5 yeartime horizon therefore entails revaluing the liabilities under theshocked mortality rates for the first 5 years but also taking account ofthe revised expectations for the subsequent run-off period.

This is shown in FIG. 23, again for a 65-year-old male, where the qx(5year shock) line represents the mortality assumptions underlying the5-year shock. During the first 5 years, the mortality rates are thefully shocked once. Beyond year 5, the mortality assumptions are basedon the revised forecast using the P-spline.

FIG. 23 also shows a 1-year shock, where the qx(1 year shock) linerepresents the mortality assumptions underlying the 5-year shock.Relative to the 5-year shock it is much more extreme: the defaultprobability of a AAA rated bond at a 1-year time horizon is much lowerthan a 5-year time horizon and so this probability corresponds to a moreextreme mortality shock. On the other hand, a 1-year shock has arelatively short-lived impact on mortality projections. After 1-year ofbad news, we (and the P-spline model) would allow for the possibilitythat this is just a temporary ‘blip’ (perhaps caused by e.g. a warmerwinter) and so the revised future expectations would be quite close tothe original best estimate. By contrast, the 10-year shock, which isrepresented by the qx(10 year shock) line, is a less extreme shock thanthe 5-year one. But because it is long-lasting, it is almost entirelyincorporated into future expectations, as can be seen from the fact thatthere is only a very modest kink at year 10.

Recalling that, according to aspects of the invention, the approach toassessing longevity capital is to ensure sufficient capital is held sothat the default probability is lower than the relevant bond class atall time horizons, the worst binding time horizon in terms of liabilityvaluations must be determined. However, it is clear from FIG. 23 that itis not possible to say ‘a-priori’ which time horizon is the worst one interms of liability valuations. In this example, and in practice, thatthe worst time horizon has been found to be generally in the region of6-8 years. However, this will vary by portfolio characteristic (thebinding time horizon for older individuals tends to be shorter than foryounger individuals). The preferred approach is to test all the relevanttime horizons for any given pension portfolio and take the most penal(i.e. worst) one.

The results of this binding time horizon testing process for thisexample are shown in Table 3, which shows the Net Present Value (NPV) ofthe shocked value of the liabilities for a confidence interval for anAAA-rated equivalent bond at different time horizons.

TABLE 3 Best Shocked liability value (AAA confidence interval) estimate5 yr 6 yr 7 yr 8 yr 9 yr 10 yr 15 yr 30 yr NPV £MM 12.10 12.729 12.73012.732 12.729 12.726 12.723 12.692 12.573 Capital NA 5.20% 5.21% 5.22%5.20% 5.17% 5.15% 4.89% 3.91%

In this example, the best estimate value of the liabilities is £12.10mM. Looking at different time horizons, the shocked liability rangesfrom £12.573 mM upwards, with the binding (i.e. worst) time horizon is 7years. In other words if sufficient assets are held to cover thisstressed liability value of £12.732, then the default probability on theliabilities is lower than that of a AAA-rated bond not only over a7-year time horizon but over all other horizons as well.

Preferably, a full re-running of the P-spline model following each shockis not conducted. P-spline modelling can be made much more flexible bytaking an approximation of the revised expected mortality ratesfollowing the shock.

Stochastic Longevity Trend Risk Quantification Method

Turning now to look at the stochastic approach to calculating longevitycapital, which builds very much on the deterministic approach describedabove. Under the stochastic approach mortality shocks are randomlysimulated using the P-spline percentiles shown in FIG. 22. For any givenstochastic simulation and any given time horizon, the experience impact(i.e. the simulated mortality rates up until the time horizon) is thenseparated out from the assumptions impact (i.e. the effect that thesimulated mortality up to the time horizon has on projected futuremortality rates). This is shown in FIG. 24, which illustratescalculations of shocked mortality rates for different time horizons fora single stochastic draw. Here, for different time horizons anyparticular simulated path is applied in full up to the time horizon andthen in part (via its effect on future expectations through re-runningthe P-spline) beyond the time horizon. While FIG. 24 illustratesmortality rates for a single age group only, it is important toappreciate that in practice, a simulation entails shocking mortalityrates across all ages (and both genders).

Having run the stochastic simulations and valued the liabilities foreach time horizon for each simulation, a probability distribution to beplotted for the liability values at each time horizon. This is shown inFIG. 25. The one-year shocks are short-lived and expectations are onlymodestly affected, giving a narrow distribution, whereas the 30 yearshocks naturally give a much wider distribution.

In accordance with aspects of the invention the required capital canthen be found by looking at the appropriate tail of the distribution.So, for example, to have a lower default probability than an equivalentAAA bond over a 1-year time horizon, for which, according to the defaultcalibration, the probability of default is 1 basis point, sufficientcapital would need to be held to cover this 1 basis point shock on thenarrow 1-year distribution. By contrast, to justify a AAA rating over a5 year time horizon, for which the probability of default is 10 basispoints, to find sufficient capital we do not need to go so far into thetail of the distribution as for the 1 year horizon, but the 5-yeardistribution itself is much wider.

As with the deterministic approach, in the stochastic approach it ishard to say a priori which time horizon will give the highest capitalrequirement. In practice, however, since this approach gives identicalcapital requirements to the deterministic approach, the binding horizonwill typically be in the range of 6-8 years.

The assessment of longevity capital for a Moody's rated product inaccordance with aspects of the present invention will now be discussed.

For a Moody's rated product, in preferred embodiments of aspects of theinvention, the approach is to hold sufficient capital to ensure that theexpected loss is lower than the Moody's idealized loss rates for thetarget debt rating. Moody's idealized loss rates are shown in Table 4.As one would expect, loss rates rise over time. In the invention, theapproach is to ensure that sufficient capital is held so that theexpected loss is lower than that of a security with the target Moody'sdebt rating at all appropriate time horizons. In this sense the capitalcalibration is conservative since the expected loss will be equal tothat of an equivalently Moody's rated security for the binding timehorizon and even lower at all other appropriate time horizons.

TABLE 4 1-Yr 2-Yr 3-Yr 4-Yr 5-Yr 6-Yr 7-Yr 8-Yr 9-Yr 10-Yr Aaa 0.0000%0.0001% 0.0004% 0.0010% 0.0016% 0.0022%  0.003% 0.0036% 0.0045% 0.0055%Aa1 0.0003% 0.0017% 0.0055% 0.0116% 0.0171% 0.0231% 0.0297% 0.0369%0.0451% 0.0550% Aa2 0.0007% 0.0044% 0.0143% 0.0259% 0.0374% 0.0490%0.0611% 0.0743% 0.0902% 0.1100% Aa3 0.0017% 0.0105% 0.0325% 0.0556%0.0781% 0.1007% 0.1249% 0.1496% 0.1799% 0.2200% A1 0.0032% 0.0204%0.0644% 0.1040% 0.1436% 0.1815% 0.2233% 0.2640% 0.3152% 0.3850% A20.0060% 0.0385% 0.1221% 0.1898% 0.2569% 0.3207% 0.3905% 0.4560% 0.5401%0.6600% A3 0.0214% 0.0825% 0.1980% 0.2970% 0.4015% 0.5005% 0.6105%0.7150% 0.8360% 0.9900%

The approach to calculating the capital requirement for any given timehorizon will now be discussed in relation to a Moody's rated product.Again, stress tests of the appropriate size (i.e. a longevity shock) areapplied to the cash flow projections and the resulting impact onliability valuations is observed. In the case of a Moody's rated productwe are essentially answering the question “How much capital do we needto hold to ensure the expected loss is no more than x %?” where x is thetarget expected loss (from the Moody's idealised loss rate table) forthe given time horizon. So, for example, at a 5 year time horizon, theexpected loss for a Aaa rated security is 0.0016%. Therefore the levelof capital that results in an expected loss of no more than 0.0016%needs to be found.

In order to estimate the expected loss for a given level of capital thevalue of liabilities at all points in the tail of the distribution ofliabilities needs to be known. This can be performed by stochasticallysimulating the Net Present Value of the liabilities. However,calculating the full distribution of the tail of this distribution istime consuming. In order to speed up our calculation, a distribution(for example, a normal distribution) is preferably fitted to the actualscheme liability distribution which produces almost identical results.Once liability distribution has been fitted to the stochasticallysimulated distribution the probability of exhausting the capital and theassociated loss for any given level of capital can be calculated.Intuitively, as the level of capital is increased, the probability ofexhausting the capital and the associated loss both decrease.

As in the approach to assessing the capital required for a Standard andPoor's and Fitch's rated product, for a Moody's product stressedmortality scenarios must be applied to the capital calculation byestimating the impact of the shocks over the relevant time horizon.

In this example, for P-spline model for a 65-year-old male, the bestestimate of the annual probability of death drops from 85 bps to 73 bpsover the first 5 years. But for a 0.1 percentile confidence intervalshock the annual probability of death drops to 70 bps. Again, this shockhas two impacts on the liability valuation: the lower mortalityexperience during those 5 years; and its effect on our assumptions aboutfuture mortality rates.

The binding time horizon which produces the worst case liabilityvaluations must then be found in order to assess the longevity capitalrequired to ensure that the estimated expected loss in that worst casescenario is no more than that of an equivalently Moody's rated security.Again, although it is not possible to say ‘a-priori’ which time horizonis the worst one in terms of liability valuations, in practice we havefound that the worst time horizon is stable for different schemeprofiles. We will check we have captured the worst case by looking atthe sensitivity to the time horizon.

The method of calculating the expected loss for assessing the longevitycapital requirement of a Moody's rated product in accordance withaspects of the invention will now be described.

In aspects of the invention the approach to calculating expected lossfor the purpose of determining the longevity risk capital requirementsis analogous to traditional Expected Loss (EL) calculations, as follows:

EL=PS×LGS

-   -   Where:        -   EL=Expected Loss        -   PS=Probability of Shortfall        -   LGS=Loss Given Shortfall

A shortfall occurs if the Net Present Value (NPV) of the actualliabilities at a given point in time exceed the sum of the ‘bestestimate’ NPV of liabilities and capital held. Thus, shortfall can beexpressed by the following equation:

Shortfall=max(0,Liab_(actual)−(Liab_(BE)+Capital))

-   -   Where:        -   Liab_(actual)=Actual NPV of liabilities        -   Liab_(BE)=Best estimate NPV of liabilities        -   Capital=Amount of capital held

For example, if the initial best estimate of liabilities is £100,capital held is £8 and the actual liabilities are £110 then theshortfall is calculated as:

Shortfall = max (0, 110 − (100 + 8))       = £2

The Probability of Shortfall (PS) is then defined as the probabilitythat a shortfall occurs. That is, the probability that the capital heldis not sufficient to cover the difference between the actual and bestestimate liabilities. PS is analogous to probability of default intraditional expected loss methodology. Thus Probability of Shortfall canbe expressed by the following equation:

PS=Prob(Liab_(actual)>(Liab_(BE)+Capital))

-   -   Where:        -   Liab_(actual)=Actual liabilities        -   Liab_(BE)=Best estimate liabilities        -   Capital=Amount of capital held

The Loss Given Shortfall (LGS) is defined as the average loss thatoccurs in the event that there is a shortfall expressed as a proportionof what would have been paid if the liability was covered in full and isanalogous to Loss Given Default (LGD) in traditional expected lossmethodology. Thus Loss Given Shortfall can be expressed by the followingequation:

LGS=shortfall/actual liabilities

In order to produce an estimated expected loss, the Probability ofShortfall and Loss Given Shortfall must be estimated. In aspects of theinvention, this is done by fitting a distribution (for example, a Normaldistribution) to estimate the actual Net Present Value of Liabilitiesover the tail region.

From the ‘fitted’ distribution, the Probability of Shortfall can then beestimated for a given level of capital by calculating the probabilitythat the actual liabilities exceed the best estimate liabilities plusthe amount of capital held.

Similarly, to estimate the Loss Given Shortfall, the tail region of the‘fitted’ distribution of liabilities can be sampled. For example, 500random draws from the tail region can be performed and then calculatethe expected loss as the average of these tail scenarios (a very largenumber of simulations is not required to achieve convergence as we arealready sampling in the tail region).

Thus, in aspects of the invention, the preferred approach to calculatingthe Expected Loss, as illustrated in FIG. 26, is as follows:

1. Fit a distribution to the scheme liabilities using an actualliability result under different longevity scenarios.2. Calculate the Probability of Shortfall (PS) from the ‘fitted’distribution, given the level of capital held.3. Calculate the Loss Given Shortfall (LGS) from the ‘fitted’distribution, given the level of capital held.4. Calculate the Expected loss as EL=PS×LGS.

Thus the Expected Loss associated with a particular longevity shock canbe calculated.

As described above, the methodologies set out above for determining thechange in the NPV of the pension scheme liabilities and the ExpectedLoss in the case of a longevity shock that is projected by thestatistical longevity projection model to occur with a certainprobability can be used to quantify longevity risk quantify and pricethe longevity risk associated with the pension scheme generally. Thiscan assist investors in understanding the longevity exposure of thefinancial instruments of the present invention.

It can also be specifically be applied to calculate the longevity riskcapital required to support the issue of a financial instrument having aspecific rating from a ratings agency.

The methodology can also be applied to calculate the size ofsubordinated tranches of capital such that they have subordinated debtratings such as BBB or Aa1, Aa2 etc. This is calculated as thedifference between the NPV of the pension cash flow liabilities for alongevity shock associated with a target rating for the tranche beingsized, and the NPV of the pension cash flows for, for example, the alongevity shock associated with the rating of the next most seniortranche of issued capital. Of course, subordinated capital may be issuedwithout a rating.

The methodology of aspects of the present invention can also be appliedgenerally to quantify the longevity risk exposure of any asset or aliability having cash flows of sums of accounts receivable and accountspayable which are dependent to some extent on the actual futuremortality experience or exposure of a group of creditors or debtors.

Ratings Method for Quantifying Longevity ‘Process’ Risk

In addition to the analysis of longevity risk and quantification of therisk capital associated therewith, an aspect of the present inventionalso provides a method for quantifying the inherent risk associated withthe process of projecting longevity for the members of a pension schemeof a certain size in the way described above. This process risk isinherent in the mortality projections for a pension scheme output fromthe statistical mortality projection model incorporating mortalitytrends in a dataset associated with reference population and alsoincorporating mortality level risk adjustments. The magnitude of theprocess risk is dependent on the size of a pension scheme membershipbeing securitized, and is particularly evident in smaller portfolios of,for example, only a few thousand members.

The risk capital required to support the process risk inherent in thecapital projections for a pension scheme of a certain size output by astatistical mortality projection model may be calculated by performing abootstrapping analysis on the reference population (such as, in the casegiven above, the CMI dataset) so as to characterise an errordistribution for the mortality projections produced by a statisticalmortality projection model. The error distribution is associated with asize of the population of the pension scheme. The characteristics of theerror distribution for the mortality of the pension scheme members, forexample the standard deviation, may be adjusted, for example by anadjustment factor, to produce an error distribution in the expected cashflows. By applying said error distribution to the Net Present Value ofthe expected cash flows, the amount of risk capital required to supportthe process risk can be quantified.

In the case of a Standard and Poor's or Fitch rated financialinstrument, the amount of risk capital to be held is calculated as theamount which is sufficient to ensure that the payment amounts on thefinancial instrument can be met in the case of a sample error in themortality projections which is projected to occur with a probability ofno more than the default probability of a bond having an equivalentrating according to the rating agency's default probability rate table.

In the case of a Moody's rated financial instrument, the amount of riskcapital to be held is calculated as the amount which is sufficient toensure that the expected loss that would result from a sample error inthe mortality projections is lower than the expected loss of a bondhaving an equivalent credit rating according to the credit ratingagency's idealised loss rate table.

The bootstrapping analysis may be performed by calculating, for N randomsamples of members of the reference population of the same size as thepopulation of the pension scheme, the mortality rate projected by thestatistical mortality projection model for that random sample for aperiod of time. By comparing each of said mortality rate projectionswith the actual mortality rate for that sample of the referencepopulation and for that period of time, the errors in the mortalityprojections can be determined and characterised. The error distributionwill generally follow a normal distribution.

An example of a bootstrapping analysis of the CMI dataset of the processrisk associated with the application of the statistical mortalityprojection method described above to a pension scheme members will nowbe described.

A series of bootstrapping analyses were carried out which comparedprojected mortality against actual mortality for 5,000 randomly sampledportfolios of members. The process of bootstrapping is as follows:

-   -   randomly select N lives from the data set    -   use the model to calculate the expected number of deaths within        the sample    -   compare the actual number of deaths in the sample with the        expected    -   repeat these steps 5,000 times for each bootstrapping analysis

In each bootstrapping analysis, the ratio of expected deaths againstactual deaths was analysed for each of the 5,000 simulations. The modelparameters used in the base case for the bootstrapping are summarised inTable 5 below.

TABLE 5 Model parameter for bootstrapping base case Model parameterSetting Fitted model dimensions Age, sex and lifestyle Amount ofhistorical data used for fitting Years 2002 to 2006 Size of portfoliofor each simulation 100,000 lives Number of bootstrapping simulations5,000 runs

FIG. 28 shows the distribution of the results from each of the 5,000simulations using the base case. In this graph, a scenario with value of100% means that the number of deaths predicted using the fittedmortality model is equal to the actual number of deaths in thatscenario.

Table 6 summarises the results of the error distribution base case; themean, standard error and 99.5^(th) percentile of the deviation betweenactual and predicted deaths were calculated.

TABLE 6 Summary of the bootstrapping output - Base Case Mean StandardScenario deviation deviation 99.5 percentile Base case 0.00% 1.02% 2.52%

The sensitivity of the of the distribution of the outcomes for differentpension scheme/sample sizes around the base case was tested for a samplesize of 50,000 lives and 100,000 lives. The results are shown in FIG. 29and Table 7.

TABLE 7 Summary of the bootstrapping output - by scheme size MeanStandard 99.5 percentile Scenario deviation deviation deviationPortfolio size 50,000 0.00% 1.46% 3.68% Portfolio size 100,000 0.00%1.02% 2.52%

Based on this analysis, it is clear that mortality level risk decreasesvery quickly as the entire portfolio exceeds 100,000 lives using a levelmortality risk model using age, sex and lifestyle.

These mortality distributions can then be adapted and used according toaspects of the present invention to quantify the risk capitalrequirement associated with process risk.

Ratings Method for Quantifying Mortality ‘Level’ Risk

As described above, the inventor's methodology provides that the initialpayment schedules of longevity financial instruments according to theinvention may be matched to the projected liabilities of a pensionscheme to its members on a given mortality basis for a referencepopulation with the mortality tables being adjusted by a mortality leveladjustment associated with each pension scheme member's socio-economiccharacteristics, even down to a post code level. Making these mortalitylevel adjustments significantly reduces the risk associated with thefinancial instrument that the mortality experience of the pension schememembership will be different from that of the reference population.

However, even after making these adjustments, there remains a degree of‘mortality level’ risk exposure in the financial instrument that themortality experience of the pension scheme membership will be differentfrom that of the level adjusted projections. This mortality level riskcan result in losses on the financial instrument that need to besupported by an associated amount of risk capital to ensure that theprobability of default of the financial instrument is below a certainlevel or to ensure that the expected loss on the financial instrument isbelow a certain level.

The inventors have therefore developed the following methodology forquantifying the mortality risk exposure of a longevity financialinstrument, and determining sufficient risk capital that needs to beheld for the financial instrument to achieve a desired credit rating.

The level risk exposure remaining in the projection model underlying thelongevity financial instrument is estimated by testing the projectionmodel on the pension scheme's existing mortality experience data. Ahistorical dataset of pension scheme member's lives on which the test isto be conducted is first defined. In this historical dataset of pensionscheme member lives, a number of deaths are known to have occurred—the‘actual deaths’, A. In the numerical example set out below, in aselected historical dataset for a pension scheme defeased by a longevityfinancial instrument according to the invention, the number of actualdeaths A that are known to have occurred is 1000.

By applying to the same historical dataset the mortality tables of anappropriate reference population (which have been adjusted to accountfor underlying trends in longevity, but not adjusted for mortality leveladjustments in the pension scheme) a number of deaths can be calculatedthat would be expected to occur if there were no mortality leveldifferences between the pension scheme membership and that of thereference population. This is the ‘expected deaths’ E. In the foregoingexample, using a longevity trend-adjusted Medium Cohort mortality tablefor the CMI assured lives reference population, the expected deaths E inthe historical dataset of pension scheme member's lives is calculated as877.

Thus, across the members of the historical pension scheme dataset, thegroup average mortality level adjustment relative to the CMI MediumCohort dataset is A/E=114.0%. Thus, on average, mortality rates of thehistorical pension scheme members are approximately 14% higher than thelives making up the CMI Medium Cohort dataset. (It should be noted that,in the inventors methodology described above, the mortality leveladjustments used in the projection of the pension scheme liabilities arecalculated on a member by member basis rather than on a group basis.)This calculation of the group average mortality level adjustment gives auseful calibration for the mortality level risk calculation, as will beseen below.

For the lives making up the historical pension scheme member dataset,there is an inherent probability distribution for the number of deathsthat occurred. This probability distribution is shown in FIG. 30 a,which shows a probability density function for the number of deaths. Inthe probability density function shown in FIG. 30 a, the 1000 actualdeaths occurred exactly at the mean of the distribution—and therefore1000 deaths was the most likely outcome. However, the probabilitydistribution of the number of deaths in the historical dataset ofpension scheme member's may very well be different from that shown inFIG. 30 a. For example, the distribution may well be as shown in FIG. 30b in which the mean deaths is 950, and the 1000 actual deaths thereforeoccurred as an event at the upper end of the distribution. Conversely,the distribution may well be as shown in FIG. 30 c in which the meandeaths is 1050, and the 1000 actual deaths therefore occurred as anevent at the lower end of the distribution.

To measure the remaining mortality risk in the projection modelunderlying the pension scheme, it is necessary to calculate theprobability density function for the number of deaths in the historicalpension scheme data on the basis of the mortality tables for thereference population adjusted to take into account a stressed mortalitylevel adjustment. These stress tests can then be used to determine astressed probability distribution that would result in the actual numberof deaths being expected to occur with a given probability. This methodis to essentially answer, by modelling the deaths in the historicaldataset, the question of where should the probability distribution shownin FIG. 30 lie to give a desired probability of the actual number ofdeaths occurring, and, specifically, what mortality level adjustmentcauses the model to give this distribution.

A stochastic approach is taken to modelling the number of deaths in thehistorical pension scheme data. The model methodology assumes that thenumber of deaths follows a binomial distribution for a given age, i.e.the distribution of deaths for members aged x is D_(x)˜Bin(N_(x),q_(x)), where N_(x) is the number of people in the dataset aged x, andq_(x) is the mortality rate for people of age x at time t. There arethen two approaches for modelling the number of deaths.

The first is a Monte Carlo simulation of the underlying binomialdistributions of the lives making up the historical dataset. By runninga large number of simulations, a probability density function of thenumber of deaths is generated.

The second is a Poisson approximation such thatD_(x)≈Poisson(N_(x),q_(x)), where D_(x) is the resulting modelledprobability distribution for age x. This provides a good approximationparticularly for N_(x)≧100 and N_(x)q_(x)≦10. Then, summing across allpossible ages, x, the Poisson approximation provides a modelledprobability density function as follows:

$D = {{\sum\limits_{x = 0}^{\omega}D_{x}} \approx {{Poisson}\left( {\sum\limits_{x = 0}^{\omega}{N_{x}q_{x}}} \right)}}$

A comparison of the two models is shown in FIG. 31 in relation to adifferent dataset to the numerical example described above. It can beseen that the two models produce very similar results. Generally, eithermodel can be used but it is preferred that the model that produces themost conservative results in a particular dataset (i.e. fewest meandeaths in the distribution) is adopted.

In continuation of the numerical example given above, by applying ineither the Monte Carlo model or the Poisson approximation model themortality tables for the reference population adjusted by a mortalitylevel adjustment of 114.0%, a probability density function should resultthat produces a mean number of deaths of 1000.

To stress the modelled probability distribution of the number of deaths,either the distribution can be merely shifted along the axis until theactual number of deaths is modelled to occur with the desiredprobability, or the model can be re-run to find the mortality leveladjustment at which the actual number of deaths is modelled to occurwith the desired probability. Whatever stress method is used, theresulting mortality level stress factor can be calculated byMean(D_(S))/E, where Mean(D_(S)) is the mean number of deaths expectedby the stressed probability distribution.

This mortality level stress factor is than applied in a recalculation ofthe projected liabilities of the pension scheme to its members used toestablish the payment schedule of the financial instrument in questionto produce a liability projection that would result from a mortalitylevel stress that is expected to occur with a specified probability.This stressed liability projection can be used to quantify the mortalitylevel risk in the pension scheme or a longevity financial instrumentundertaking to make payments mirroring the pension scheme's liabilities.By this quantification method, the risk capital requirement needed to beheld to support the longevity financial instrument in the case of amortality level shock determined to occur with a specified probabilitycan also be calculated. This quantification can be done using a Value atRisk method or another suitable method.

Similarly, the stressed liability projection method can also be used todetermine the risk capital requirement needed to be held to achieve fora longevity financial instrument a desired rating from a rating agency(at least in relation to mortality level risk).

This risk rating method will be applied in continuation of the numericalexample set out above to achieve an ‘A’ rating from Standard & Poor's ona longevity financial instrument undertaking to make payments mirroringthe pension scheme's liabilities. To determine the mortality level riskcapital requirement needed to achieve a Standard and Poor's ‘A’ rating,the stressed probability distribution must be determined that gives theprobability of the actual number of deaths occurring as being equal tothe probability of default of an ‘A’ rated bond according to Standard &Poor's ratings table for an appropriate time horizon, t:

Pr(Number of deaths>Actual deaths)≦S&P PD(rating,tenor)

For an S&P ‘A’ rated bond at a 14 year time horizon, the probability ofdefault is 3.40%.

Stressing the Poisson approximation model applied to the historicalpension scheme data, the resulting probability distribution that givesan expected number of deaths as being 1000 or more with a probability of3.40% provides a stressed mean number of deaths of Mean(D_(S))=944.Performing the same stress test on the Monte Carlo simulation of deathsin the historical pension scheme data, the stressed mean number ofdeaths in that dataset of Mean(D_(S))=951.

Taking the Poisson approximation model as the most conservative, theresulting mortality level stress factor is Mean(D_(s))/E=944/877=107.6%.Thus the mortality level adjustment that is modelled to occur with aprobability equal to the desired default probability is 107.6%.

By recalculating the projected liabilities of the pension scheme to itsmembers on the basis of this ‘group’ mortality level adjustment for allcurrent pension scheme members, a stressed liability projection can beobtained. The mortality level risk capital requirement to cover theliabilities up to this mortality level stress can then be calculated by,for example, calculating the difference in the present value of thestressed and the ‘best estimate’ liability projections.

To obtain a Moody's rating, a similar stress testing method should befollowed to determine the mortality level risk capital than needs to beheld to ensure that the expected losses on the longevity financialinstrument are no more than a desired percentage.

Method of Integrating Longevity Risk with Market Risk

The various risk sources inherent in the longevity financial instrumentsof the inventor's methodology each contribute to a risk capitalrequirement to support the financial instrument and may combine toproduce a diversification benefit that effectively reduces the totalrisk capital requirement. By determining this diversification benefit,the resulting reduction in risk capital can be gained from by reducingthe amount of risk financing required to support the longevity financialinstrument.

The diversification benefit of the various risk sources in the longevityfinancial instruments can be determined by the following two approaches:the variance-covariance method, and the Monte Carlo simulation method.

The Variance-Covariance Method

Consider a simple example where the longevity financial instrument isexposed to 3 different risk sources A, B and C (which could be longevitytrend, mortality level, and credit risk, say). The correlationcoefficients, ρ_(ij), between the different risk source i, j, must thenbe determined. These correlation coefficients may be assumed to take acertain value. Estimation and modelling may be used to build acovariance matrix. In this example the correlation coefficients betweeneach individual risk are taken to be as set out in the covariance matrixshown in Table 8.

TABLE 8 Risk A B C A 1 0.25 0.5 B 1 0.5 C 1

Given these assumed correlations, the total diversified capitalrequirement is then calculated from the individual risk capitalcomponents C_(i) for each risk source i by applying the ‘square root ofthe sum of squares rule’:

C _(total)=√{square root over (ΣC _(i) ²+Σρ_(ij) C _(i) C _(j))}

wherein C_(total) is the total diversified risk capital requirement.

Taking the undiversified individual risk capital requirements C_(i) forrisk sources A, B and C as £10, £20 and £50, respective, the totalundiversified risk capital requirement is then £80 (i.e. a mere sum ofthe C). However, using the ‘square root of the sum of squares rule’ inconjunction with the assumed covariance matrix for the risk sources ishown in Table 8, the total diversified risk capital requirementC_(total) is calculated to be £68.

Thus in this example, the diversification benefit of the various risksources is determined to be £12. Thus the total risk capital requirementcan be reduced by an equivalent amount without increasing the overallrisk exposure associated with the longevity financial instrument.

The Monte Carlo Simulation Method

In certain situations, the Monte Carlo simulation method is used as analternative method to calculate the total risk capital required to coverrisks from a number of risk sources, i. The example below sets out thismethod to calculate the diversification benefit resulting from bothcombined credit risk and longevity trend risk exposure.

To calculate the capital requirement due to credit risk in isolation, aMonte Carlo simulation is used for bond defaults. This is run over alarge number of scenarios (say 10,000 runs). Each scenario gives a valuefor the credit losses as a result of the bond defaults over theparticular scenario. By considering all 10,000 scenarios, a distributionof credit losses can be constructed, and this can be used to calculatethe capital that would be sufficient to cover credit losses to aparticular level of certainty (or probability). This probability wouldbe calibrated to a particular rating agency notch (or alternativemeasure) by considering the x-percentile point on the credit lossdistribution.

Similarly, to calculate the capital requirement due to longevity trendrisk in isolation, a Monte Carlo simulation of future mortalityimprovement rates is again used (also for, say, 10,000 runs). This isrun over a large number of scenarios, which is then used to calculatethe capital requirement for longevity trend risk in the same way as isdone above for credit risk.

The above analysis gives separate risk capital values for credit andlongevity risk considered in isolation. However, in order to calculatethe total diversified risk capital requirement that needs to be heldwhen there is a combined exposure to both of these risks (i.e. a totalincorporating the diversification benefit), the following method isused.

It is first assumed that combined credit risk and longevity trend riskare uncorrelated, i.e. that the correlation coefficient between them iszero. The (unordered) random scenarios from each Monte Carlo model isthen combined. By summing the combined credit loss and the longevitytrend loss for each combined pair of scenarios, the total losses for anew (combined) credit and longevity scenario is provided. By consideringall 10,000 combined scenarios, a distribution of the aggregate creditand longevity trend losses can be constructed.

Given the combined loss distribution, the total diversified risk capitalrequirement can be calculated by taking the x-percentile point on thedistribution and holding capital to cover this eventuality.

The following is a simplified numerical example of the Monte Carlomethod in which there have been modelled 20 credit and longevityscenarios. The individual losses for the credit and longevity scenariosare shown in Table 9 in the first two columns, and the aggregate lossesfor the combined risks are shown in the last column.

TABLE 9 Credit losses Longevity losses Total combined losses Scenario17.3 −3.0 4.3 Scenario2 −11.6 2.6 −9.1 Scenario3 7.2 −1.3 5.9 Scenario4−9.8 −5.2 −15.1 Scenario5 −28.9 −1.1 −30.0 Scenario6 9.5 −11.8 −2.4Scenario7 21.1 0.1 21.3 Scenario8 −17.5 −2.8 −20.2 Scenario9 12.9 −1.311.5 Scenario10 −20.6 1.2 −19.5 Scenario11 −23.4 −8.2 −31.6 Scenario126.2 −1.6 4.6 Scenario13 26.0 −0.5 25.5 Scenario14 −45.9 1.2 −44.7Scenario15 −13.5 −1.7 −15.2 Scenario16 −2.2 −6.6 −8.8 Scenario17 15.7−3.8 11.9 Scenario18 31.7 9.7 41.4 Scenario19 −24.6 3.5 −21.1 Scenario204.9 −5.4 −0.5 90% percentile 21.6 2.7 21.7

By calibrating to the 90^(th)-percentile point on each lossdistribution, the undiversified individual risk capital requirements forcredit risk and longevity risk are £21.6 and £2.7, respectively. Thetotal undiversified risk capital requirement is then £24.3.

However, taking the 90^(th)-percentile point on the distribution for thetotal combined losses for each scenario, the modelled the totaldiversified risk capital requirement is calculated to be £21.7.

Thus in this example, the diversification benefit of the various risksources is determined as £2.6. Thus the total risk capital requirementcan be reduced by an equivalent amount without increasing the overallrisk exposure associated with the longevity financial instrument in thetotal risk

Providing Risk Capital and Supporting the Issuance of LongevityFinancial Instruments

The risk capital can be held in the form of subordinated tranches ofdebt and equity, issued in the form of, for example, capital notes andequity notes. Due to the low volatility in longevity risk of, forexample a pension scheme membership, the opportunity presented toinvestors to create value from subordinated notes exposed to thislongevity risk is rather limited and is confined to the tail of thedistribution. This may limit interest from investors and also createpotential barriers to achieving an underwriting of the longevity risk inthe capital markets. According to aspects of the present invention, toincrease the opportunity for investors to create value by investing inthese subordinated tranches of capital, and to make them moreattractive, the subordinated capital may comprise exposure to longevityrisk and to asset risk together. Thus the subordinated capital issuedaccording to this aspect of the invention will support the risk exposureof the senior product to longevity risk and will also support the riskexposure of the assets underlying the issue of the senior product.

A financial instrument according to the present invention may be issuedwhere it is not underwritten or is self-underwritten (i.e. where thecorporate sponsor of the pension scheme invests in the subordinated riskcapital in order to support the issue of the financial product). Thismay occur where, for example, the value of a pension scheme'sliabilities is so large that there is not the underwriting capacity inthe market available to support the issue of a capital markets productaimed at securitizing the longevity risk of the pension scheme. In thisinstance, the operation of the pension scheme may be transferred ontothe risk management system platform and a financial product according toaspects of the present invention may be issued while the pension schemesponsor provides the risk capital to support the issue. The subordinatedcapital then held by the sponsor may later be sold on by the sponsor.

Due to the fact that the entities issuing the various tranches offinancial instruments according to aspects of the invention will alwaysoperate on the basis of their funding duration always exceeding theirasset duration, this will be a benefit to potential capital noteinvestors who will be able to access exposure to longevity with anenhanced yield provided by the additional exposure to the assetportfolio. For many traditional leveraged credit investors, this willprovide an attractive new alternative way of achieving leveragedexposure to credit, without the need to additionally expose themselvesto the risks associated with refinancing of short term debt and mark tomarket models, which apply to leveraged investment models whichnegatively mis-match the duration of their assets and liabilities byborrowing short and lending long.

By transferring the operation of the pension scheme onto the riskmanagement system supporting the methods of aspects of the presentinvention, the risk management system provides a powerful tool enablingthe careful and calculated management of the liabilities of the pensionscheme. By the capital projection modelling methods of aspects of thepresent invention, pension scheme trustees or corporate sponsors may usethe risk management system to analyse the costs associated with thesecuritization of the cash flows of liabilities to individual pensionscheme members and take any appropriate action to manage thoseliabilities. For example, the trustees of a pension scheme or thecorporate sponsor thereof may identify, using the risk managementsystem, a number of deferred pension members for whom the cost ofinvesting in a financial instrument according to aspects of the presentinvention to securitize that members liabilities is particularly costly,at, for example, £100,000 each. Having this information, the trustees orthe corporate sponsor may decide to manage those liabilities by offeringthose members a cash incentive of, for example, £80,000, to transfer outof the pension scheme. This capability for liability management in thisway is provided by the risk management system and methods of aspects ofthe present invention.

The Longevity Capital Model

The Longevity Capital Model (LCM) for cash flow projection will now bedescribed.

The LCM is a cash flow projection model in accordance with aspects ofthe invention that carries out member-by-member pension cash flowprojection and valuation. FIG. 26 illustrates the main elements of thismodel.

The input sheets contain member-by-member information on factors whichdrive the member's pension entitlement such as accrued pensionentitlement, as well as factors driving the member's expected mortalitysuch as age and gender. The sheets also contain pension scheme levelinformation such as the rules surrounding indexation of the variousslices of benefits before and during retirement.

The member state model estimates the likelihood of a given member beingalive or deceased (and if deceased whether their spouse is alive ordeceased) on a given date. This probability projection is based on themortality assumptions derived using the P-spline and fed into the modelas an input.

The benefit calculator estimates the pension cash flow to be paid to agiven member in a given period on the assumption that they are alive inthat period. So, for example, it calculates the pension cash flow if themain member is still alive and also the pension cash flow if they aredead but the spouse is still alive. Beyond this, it calculates theseparate ‘slices’ of benefits—so, for example, it calculates a member'scontracted out benefits separately from the standard pension benefits,taking account of different indexation requirements for each.

Finally, the aggregation section of the model draws together the memberstate model and the benefit calculator. By taking account of theprobability of paying each type of pension benefit in each period aswell as the size of that benefit, the model calculates expected cashflows. Net Present Values are derived based off swap rates for fixedcash flows and index-linked curves for indexed cash flows. Longevityrisk capital requirements are then derived using one of the approachesdescribed above to apply specific shocks to the mortality assumptions.

There will now be described a number of exemplary Pension DefeasanceSecurities products, which can be offered to pension scheme trustees andcorporate sponsors and used to immunize a pension scheme from longevityrisk by at least partially defeasing the pension scheme for at least apredetermined period.

The Buyout Equivalent Bond, or ‘Blue Bond’

This is economically equivalent to a buyout and therefore the mostcomprehensive product, which pays cash flows that mirror the actualliabilities of the scheme to its members. This is achieved by using theproprietary risk management systems to analyse the pension schememembership data and scheme rules to create a projection of expectedliabilities. Payments on this bond will fully reflect all relevantpension scheme legislation including Barber adjustments, GMP step ups,and anti-franking legislation.

The Blue Bond, although economically equivalent to buyout, fundamentallydiffers from existing insurance buyout solutions as it is designed to beheld as an asset of the pension scheme, under the control of thescheme's existing trustees. As with all products according to aspects ofthe invention, the Blue Bond is primarily designed for use by ongoingschemes. However, if required, it could also be structured to provide afull buyout solution for a closed pension scheme from which the sponsorwishes to be de-linked.

Once a pension scheme has bought a Blue Bond and the scheme data andrules are on the administrative platform of the risk management system,it is then very easy to price additional tranches of benefits,additional accruals, or increased compensation. Additional tranches ofbenefits can then be purchased at a defined price, making the financialimpact of running a defined benefit scheme transparent to the sponsor.

The Term Buyout Bond, or ‘Term Blue Bond’

This product pays cash flows that mirror the actual liabilities of thepension scheme to its members for a defined period. The product is idealfor pension schemes which are seeking to immunise a significant part oftheir risk, but may not have the resources to totally defease theliability. The Term Blue Bond allows a pension scheme to choose theperiod of risk that it covers, based upon its resources and riskappetite.

This product is likely to be popular with schemes that are looking tomove to a position of full funding and total risk removal over a periodof time as they will be able to reduce risk and volatility significantlyand then extend the horizon of cover as they receive additionalcontributions from the sponsor or surplus is generated from exposure tohigher risk assets.

The Deferred Payment Bond, or ‘Geared Blue Bond’

This product is designed for schemes that are not currently fully fundedand cannot, therefore, buy a full Blue Bond. This product provides fullimmunisation of risk for the life of the scheme, with part of the costpayable over a number of years. This makes it easier for the sponsor tocover the cost of filling the deficit in a phased way, while putting thetrustees in a fully defeased position and fully removing the deficitvolatility.

It could also be of value to schemes which could buy a full Blue Bondoutright but choose to retain some non-matching assets within the schemeto try and achieve extra return, which could then be used to grantdiscretionary benefits or reduce sponsor contributions in respect offuture accrual.

The Pro-Rata Bond, or ‘Light Blue Bond’

This is a Blue Bond that pays out a defined percentage of schemebenefits for the full term of the scheme. Alternatively, payments on thebond may be linked with the liabilities of the pension scheme to anydefined segment of its members, such as, for example, males or femalesonly, members over a certain age, etc.

This product allows a scheme to choose exactly what proportion orsegment of its liabilities it wishes to cover. The use of this bond isvery flexible as it can be used to replace a traditional bond portfoliowith an investment that mirrors the inflation sensitivity, duration,embedded options and longevity of the scheme's actual liabilities. Itcan also be used as part of a dynamic investment strategy to graduallymove towards a complete removal of financial risk from the pensionscheme as the proportion of the liabilities that are covered by the bondis increased.

The Term Deficit Volatility Removal Bond, or ‘Green Bond’

This product is designed for pension scheme sponsors, which areconcerned about deficit volatility stemming from IAS 19 and FRS 17. Todeal with this issue, a number of different solutions are availabledepending on the requirements.

A typical example would involve transfer of the scheme assets to theissuer of the financial instrument which would undertake to pay all ofthe benefits due to members for 10 years, at the end of which the issuerof the financial instrument would return to the scheme an amount thatguarantees the IAS19 surplus/deficit to a pre-specified level. As aresult, the sponsor would be protected against deficit volatility forthe life of the investment.

The Buyout Equivalent Fixed Inflation Bond, or ‘White Bond’

This product is the same as the full Blue Bond except that it pays onthe assumption that there is no future variability in inflation, i.e. itis priced on the basis of a fixed inflation assumption. The purpose ofthis product is to provide longevity cover to schemes who may havealready removed their exposure to variable inflation through thederivative markets. This product may also be suitable for schemes wherethe sponsor is comfortable with the inflation risk—e.g. when a companyhas an income stream which is linked to inflation—but wishes to hedgeexposure to longevity.

It will be understood that many other Pension Defeasance Securitiesproducts fall within the scope of the invention and those bondsdescribed above are presented herein only as an example. In particular,bonds and other suitable securities and derivatives can be structured tomeet the specific objectives of a pension scheme according to thescheme's rules, membership, appetite for risk and available resources.This can be achieved by analyzing each of the separate risks the pensionscheme faces, down to the individual member level, and removing thoseexposures the scheme does not wish to manage, whilst retaining thosewith which the scheme is comfortable and wishes to retain the upsidepotential. Bonds and other suitable securities and derivatives cantherefore be issued which are capable of providing risk specific orpartial defeasance or the total elimination all scheme risks, up to abuyout level.

The Life Expectancy Bond, or Purple Bond

This product pays cash flows that reflect actual liabilities of a schemeto its members subject to an agreed age limit for each member or definedgroup of members (group defined by reference to age, gender, status(deferred/pensioner) etc. The Purple Bond can therefore be used toprovide cost effective risk management for scheme sponsors and trusteeswho do not want to pay excessive premiums for risks they consider to beof low probability.

The Best Estimate Cash Flows Only Bond, or Red Bond

The Red Bond pays cash flows that reflect projected liabilities of ascheme to its members at time of issue based upon longevity parametersrequired by the trustees and sponsor (these cash flows may reflect bestestimate longevity or may be increased or decreased to meet additionalor reduced risk coverage requirements); its cash flows are not subjectto adjustment by reference to actual mortality outcomes but are subjectto adjustment for all other factors (inflation and member discretionssuch as cash commutation, transfers out etc).

It will be appreciated that in putting into effect any embodiments ofthe invention, any or all calculations may be carried out by dataprocessing apparatus having processing means, memory means, data inputmeans and data output means, using suitable software which may begeneric or specifically designed for use in the context of the presentinvention.

1. A computer-implemented method of estimating a capital reserverequirement to cover the longevity risk exposure of a financialinstrument in the case of a future longevity shock, the financialinstrument undertaking to pay to an investor sums according to a paymentschedule of amounts arranged to match with the future cash flowobligations of a pension scheme to at least a portion of its members;the method comprising: (a) calculating, using computing apparatus, anexpected payment schedule of the financial instrument by calculatingwhat the cash flow obligations of the pension scheme to its relevantmembers would be in the case of an expected longevity scenario for thepension scheme membership occurring; (b) calculating, using computingapparatus, a present value of the financial instrument in the case of astressed longevity scenario for the pension scheme membership in which alongevity-related shock to the expected longevity scenario of thepension scheme membership occurs; and (c) calculating, using computingapparatus and using the calculations of the expected payment scheduleand a present value of the financial instrument in the case of astressed longevity scenario, an estimate of the longevity capitalreserve required to ensure that the future cash flow obligations of thefinancial instrument would be covered in the event that the stressedlongevity scenario were to occur.
 2. A method as claimed in claim 1,wherein the longevity-related shock is taken to be a mortality trendshock to the expected longevity scenario, such that the longevitycapital reserve requirement estimated by the model is that required tocover the longevity risk exposure of the financial instrument in thecase of a future mortality trend shock occurring.
 3. A method as claimedin claim 2, wherein step (b) comprises: calculating, using computingapparatus, a stressed payment schedule of the financial instrument bycalculating what the cash flow obligations of the pension scheme wouldbe in the case of a stressed longevity scenario for the pension schememembership in which a mortality trend shock to the expected longevityscenario of the pension scheme membership occurs; and calculating, usingcomputing apparatus, a present value of the financial instrument in thecase of the stressed payment schedule occurring.
 4. A method as claimedin claim 3, wherein, in step (b), the stressed longevity scenario isselected on the basis of a longevity scenario incorporating a givenmortality trend shock, wherein, in step (c), the estimate of thelongevity capital reserve requirement is calculated as the differencebetween a present value of the stressed payment schedule and a presentvalue of the expected payment schedule.
 5. A method as claimed in claim3, wherein step (a) comprises: modelling, using computing apparatus, thelongevity of the pension scheme membership to produce a longevityprojection model for the pension scheme which provides longevityprojections for the pension scheme membership; and selecting theexpected longevity scenario as the best estimate longevity projection ofthe pension scheme membership provided by the longevity projectionmodel.
 6. A method as claimed in claim 5, wherein the longevityprojection model takes into account mortality tables for a suitablereference population adjusted for: at a group level, underlyingmortality trends modelled in the suitable reference population; and/orat an individual level, mortality level adjustments corresponding toeach pension scheme member's socio-economic characteristics.
 7. A methodas claimed in claim 5, wherein, in step (b), the stressed longevityscenario is constructed to comprise: over the duration of a given timehorizon, a shocked longevity projection of the pension scheme membershipincorporating a mortality trend shock; and after the given time horizon,a best estimate longevity projection of the pension scheme membershipoutput by the longevity projection model, the model having been updatedto take into account the experience of the mortality trend shock overthe duration of the time horizon.
 8. A method as claimed in claim 7,wherein the longevity projection model further provides confidenceintervals around the best estimate longevity projection by which theprobability of a longevity projection, incorporating a mortality trendshock given by the confidence intervals, occurring can be estimated, andwherein, in step (b), the shocked longevity projection of the stressedlongevity scenario is selected on the basis of a confidence intervallongevity projection of the longevity projection model, the shockedlongevity projection being projected to occur with a probabilityestimated according to the longevity projection model.
 9. A method asclaimed in claim 8, wherein step (b) includes selecting the shockedlongevity projection as being that longevity projected to occur by thelongevity projection model with a given probability.
 10. A method asclaimed in claim 9, wherein step (c) comprises calculating the estimateof the longevity capital reserve requirement as the difference between apresent value of the stressed payment schedule and a present value ofthe expected payment schedule.
 11. A method as claimed in claim 9,wherein the stressed longevity scenario is calibrated to a probabilitygiven by a rating agency's default probability rate table, whereby theshocked longevity projection of the stressed longevity scenario isselected to be that projected by the longevity projection model to occurwith a probability of no more than the default probability of acorporate bond having a given rating according to the defaultprobability rate table.
 12. A method as claimed in claim 11, whereinstep (c) comprises calculating the longevity capital reserve requirementas the difference between a present value of the stressed paymentschedule and a present value of the expected payment schedule.
 13. Amethod as claimed in claim 11, further comprising: repeating step (b) toperform a plurality of separate stress tests each providing a stressedpayment schedule on the basis of a different stressed longevityscenario, each different stressed longevity scenario comprising ashocked longevity projection incorporating a mortality trend shock overone of a plurality of different time horizons, the shocked longevityprojection of each stressed longevity scenario being calibrated to theprobability given by the rating agency's default probability rate tablefor the same rating corporate bond and for a tenor matching the relevanttime horizon for the relevant mortality trend shock; wherein step (c)comprises: calculating a present value of each of the plurality ofstressed payment schedules; identifying the worst stressed paymentschedule having the highest present value; and calculating the estimateof the longevity capital reserve requirement as the difference betweenthe present value of the worst stressed payment schedule and a presentvalue of the expected payment schedule.
 14. A method as claimed in claim8, wherein step (b) includes repeatedly stochastically simulating astressed payment schedule for the financial instrument in whichsimulation a sequence of random mortality trend shocks are applied atgiven time horizons, each mortality trend shock being applied byrandomly selecting, at a given time horizon, the shocked longevityprojection in accordance with a probability distribution for the pensionscheme membership longevity projection modelled by the confidenceintervals of the longevity projection provided by the longevityprojection model, the repeated simulations providing, for each giventime horizon, a plurality of simulated stressed payment scheduleoutcomes each being associated with a randomly stressed longevityscenario.
 15. A method as claimed in claim 14, wherein step (c)comprises calculating, for each given time horizon, a present value ofeach of the plurality of simulated stressed payment schedule outcomes,and generating, for each given time horizon, a probability distributionof present values of stressed payment schedules.
 16. A method as claimedin claim 15, wherein step (c) comprises: determining, for a given timehorizon, a present value of a stressed payment schedule which, from theprobability distribution of present values of stressed payment schedulesfor that time horizon, is simulated to occur with a given probability;and calculating the estimate of the longevity capital reserverequirement as the difference between the present value of the stressedpayment schedule that is simulated to occur with the given probabilityand a present value of the expected payment schedule.
 17. A method asclaimed in claim 15, wherein, in step (c), the calculation of theestimate of the longevity capital reserve requirement is calibrated to aprobability given by a rating agency's default probability rate table,whereby step (c) comprises: determining, for a given time horizon, apresent value of a stressed payment schedule which, from the probabilitydistribution of present values of stressed payment schedules for thattime horizon, is simulated to occur with a probability of no more thanthe default probability of a corporate bond having a given ratingaccording to the default probability rate table for a tenor matching therelevant time horizon; wherein step (c) further comprises: calculatingthe estimate of the longevity capital reserve requirement as thedifference between the present value of the stressed payment schedulesthat is simulated to occur with the given probability and a presentvalue of the expected payment schedule.
 18. A method as claimed in claim15, wherein, in step (c), the calculation of the estimate of thelongevity capital reserve requirement is calibrated to a probabilitygiven by a rating agency's default probability rate table, whereby step(c) comprises: determining, for each time horizon, a present value of astressed payment schedule which, from the respective probabilitydistribution of present values of stressed payment schedules for thattime horizon, is simulated to occur with a probability of no more thanthe default probability of a corporate bond having a given ratingaccording to the default probability rate table for a tenor matching therelevant time horizon; selecting, from across all time horizons, theworst present value of the stressed payment schedules that is simulatedto occur with the given probability; and calculating the longevitycapital reserve requirement as the difference between the worst presentvalue of the stressed payment schedules and a present value of theexpected payment schedule.
 19. A method as claimed in claim 15, whereinstep (c) comprises: calculating, on the basis of the probabilitydistribution of present values of stressed payment schedules for a giventime horizon, the amount of longevity capital reserve required to ensurethat the expected loss on the financial instrument for that time horizonis no greater than a given expected loss.
 20. A method as claimed inclaim 19, wherein the longevity capital reserve requirement calculationis calibrated to a rating agency's idealized loss rate table, wherebythe given expected loss is an expected loss given by the idealized lossrate table.
 21. A method as claimed in claim 20, wherein the calculationof the longevity capital reserve requirement includes calculating, for agiven amount of capital held, the expected loss on the financialinstrument, the calculation of the expected loss comprising: fitting amodel distribution to the probability distribution of present values ofstressed payment schedules for the given time horizon; calculating, fromthe fitted distribution, a probability of a shortfall occurring giventhe amount of capital held; and calculating, from the fitteddistribution, an estimate of the average shortfall given the amount ofthe capital held and calculating the loss given shortfall as the ratioof the average shortfall to the present value of the expected paymentschedule; the calculation of the expected loss including multiplying theprobability of a shortfall by the loss given shortfall.
 22. A method asclaimed in claim 21, wherein the calculation of the longevity capitalreserve requirement includes iteratively performing calculations of theexpected loss for different given amounts of capital held to determinethe amount of capital required to ensure that the expected loss is notless than the given expected loss.
 23. A method as claimed in claim 21,wherein the calculation of the probability of a shortfall occurringincludes calculating, from the fitted distribution, the probability thatthe present value of the stressed payment schedule will exceed thepresent value of the expected payment schedule by the given amount ofcapital held; and wherein the calculation of the average shortfall giventhe amount of capital held includes taking an average of a random sampleof present values in the tail of the fitted distribution in the regionof present values greater than the sum of the present value of theexpected payment schedule by and the given amount of capital held.
 24. Amethod as claimed in claim 1, wherein the method is used to calculate anestimate of the size of difference tranches of longevity risk capitalassociated with the financial instrument.
 25. A method as claimed inclaim 1, wherein the longevity-related shock is taken to be a shockresulting from a sample error resulting from the size of the pensionscheme membership, such that the longevity capital reserve requirementestimated by the method is that required to cover the longevity riskexposure of the financial instrument in the case of the longevity of thepension scheme membership being different from the expected longevityscenario due to a sample error occurring.
 26. A method as claimed inclaim 25, wherein step (b) comprises: characterising, by performing abootstrapping analysis on a reference population using data processingapparatus, a sampling error distribution for the longevity projectionsaccording to an expected longevity scenario for the referencepopulation, the sampling error distribution being associated with a sizeof the population of the pension scheme; and applying, using dataprocessing apparatus, said sampling error distribution to a presentvalue of the expected cash flows to produce a distribution of presentvalues of the financial instrument occurring in the case of a sampleerror; and wherein step (c) comprises determining the amount oflongevity risk capital required to ensure that: the payment schedule onthe financial instrument would be covered in the case of a sample errorin the longevity projections which is projected to occur by thebootstrapping analysis with a given probability; or the expected lossthat would result from a sample error in the longevity projections islower than a given expected loss.
 27. A method as claimed in claim 26,wherein: the given probability is the default probability of a bondhaving an equivalent rating according to a rating agency's defaultprobability rate table; or wherein the given expected loss is theexpected loss of a bond having an equivalent rating according to arating agency's idealised loss rate table.
 28. A method as claimed inclaim 26, wherein said bootstrapping analysis comprises: calculating,for N random samples of members of the reference population of the samesize as the population of the pension scheme, the projected mortalityrate according to the expected longevity scenario for the referencepopulation for that random sample for a period of time; comparing, foreach sample, the projected mortality rate with the actual mortality ratefor that sample of the reference population and for that period of timeto determine errors in the mortality rate projections; andcharacterising the distribution of the errors in the longevityprojections of the longevity projection model.
 29. A method as claimedin claim 1, wherein the longevity-related shock is taken to be amortality level shock to the expected longevity scenario of the pensionscheme membership, such that the longevity capital reserve requirementestimated by the model is that required to cover the mortality levelrisk exposure in the financial instrument in the case of a futuremortality level shock occurring.
 30. A method as claimed in claim 29,wherein step (b) comprises: determining the actual number of deathsalready occurred in an available historical data set for the pensionscheme; modelling, for the historical data set and using data processingapparatus, a probability density function for the number of deaths inthe historical data set based on the mortality tables for a suitablereference population; determining, using data processing apparatus, ashocked mortality level adjustment which, when applied to the mortalitytables in the said model, provides a probability density function forwhich the probability of the actual number of deaths occurring is equalto a given probability, the stressed longevity scenario beingconstructed by applying the shocked mortality level adjustment to themortality tables; and calculating, using data processing apparatus, astressed payment schedule of the financial instrument by calculatingwhat the cash flow obligations of the pension scheme would be in thecase of the stressed longevity scenario occurring and calculating thepresent value of the financial instrument in the stressed longevityscenario on the basis of the stressed payment schedule.
 31. A method asclaimed in claim 30, wherein the shocked mortality level adjustment iscalibrated to a probability given by a rating agency's defaultprobability rate table such that the actual number of deaths isprojected to occur with a probability of no more than the defaultprobability of a corporate bond having a given rating according to thedefault probability rate table.
 32. A method as claimed in claim 30, inwhich the model of the probability density function of scheme memberdeaths assumes that deaths follow a binomial distribution for a givenage.
 33. A method as claimed in claim 32, wherein the model of theprobability density function of scheme member deaths is based on aPoisson approximation.
 34. A method as claimed in claim 32, wherein themodel of the probability density function of scheme member deaths isbased on a Monte Carlo simulation.
 35. A method as claimed in claim 29,wherein step (b) comprises: determining the actual number of deathsalready occurred in the available historical data for the pensionscheme; modelling, for the historical data set and using data processingapparatus, a probability density function for the number of deaths inthe data set based on the mortality tables for a suitable referencepopulation; and calculating, using data processing apparatus, the valueat risk in a present value of the payment schedule amounts of thefinancial instrument due to a shocked mortality level adjustment forwhich, when applied to the expected longevity scenario for the pensionscheme membership, the actual level of deaths in the historical data ismodelled to occur with a given probability, the value at risk being usedin step (c) to determine the longevity capital reserve requirement. 36.A method as claimed in claim 35, in which the model of the probabilitydensity function of scheme member deaths assumes that deaths follow abinomial distribution for a given age.
 37. A method as claimed in claim36, wherein the model of the probability density function of schememember deaths is based on a Poisson approximation.
 38. A method asclaimed in claim 36, wherein the model of the probability densityfunction of scheme member deaths is based on a Monte Carlo simulation.39. A computer-implemented method of estimating a capital reserverequirement to cover the longevity risk exposure of a financialinstrument in the case of a future mortality trend shock, the financialinstrument undertaking to pay to an investor sums according to a paymentschedule of amounts arranged to match with the future cash flowobligations of a pension scheme to at least a portion of its members;the method comprising: (a) calculating, using computing apparatus, anexpected payment schedule of the financial instrument by calculatingwhat the cash flow obligations of the pension scheme to its relevantmembers would be in the case of an expected longevity scenario for thepension scheme membership occurring; (b) calculating, using computingapparatus, a stressed payment schedule of the financial instrument bycalculating what the cash flow obligations of the pension scheme wouldbe in the case of a stressed longevity scenario for the pension schememembership in which a mortality trend shock to the expected longevityscenario of the pension scheme membership occurs; and (c) calculating,using computing apparatus and using the calculations of the expectedpayment schedule and the stressed payment schedule, an estimate of thelongevity capital reserve required to ensure that the future cash flowobligations of the financial instrument would be covered in the eventthat the stressed longevity scenario were to occur.
 40. Acomputer-implemented method of estimating a capital reserve requirementto cover the longevity risk exposure of a financial instrument in thecase of a future mortality level shock, the financial instrumentundertaking to pay to an investor sums according to a payment scheduleof amounts arranged to match with the future cash flow obligations of apension scheme to at least a portion of its members; the methodcomprising: (a) calculating, using computing apparatus, an expectedpayment schedule of the financial instrument by calculating what thecash flow obligations of the pension scheme to its relevant memberswould be in the case of an expected longevity scenario for the pensionscheme membership occurring; (b) calculating, using computing apparatus,a present value of the financial instrument in the case of a stressedlongevity scenario for the pension scheme membership in which amortality level shock to the expected longevity scenario of the pensionscheme membership occurs; and (c) calculating, using computing apparatusand using the calculations of the expected payment schedule and apresent value of the financial instrument in the case of a stressedlongevity scenario, an estimate of the longevity capital reserverequired to ensure that the future cash flow obligations of thefinancial instrument would be covered in the event that the stressedlongevity scenario were to occur.
 41. A computer-implemented method ofestimating a capital reserve requirement to cover the longevity riskexposure of a financial instrument in the case of a sample error, thefinancial instrument undertaking to pay to an investor sums according toa payment schedule of amounts arranged to match with the future cashflow obligations of a pension scheme to at least a portion of itsmembers, the sample error resulting from the size of the pension schememembership; the method comprising: (a) calculating, using computingapparatus, an expected payment schedule of the financial instrument bycalculating what the cash flow obligations of the pension scheme to itsrelevant members would be in the case of an expected longevity scenariofor the pension scheme membership occurring; (b) calculating, usingcomputing apparatus, a present value of the financial instrument in thecase of a stressed longevity scenario for the pension scheme membershipin which a sample error resulting from the size of the pension schememembership occurs; and (c) calculating, using computing apparatus andusing the calculations of the expected payment schedule and a presentvalue of the financial instrument in the case of a stressed longevityscenario, an estimate of the longevity capital reserve required toensure that the future cash flow obligations of the financial instrumentwould be covered in the event that the stressed longevity scenario wereto occur.
 42. A computer-implemented method of calculating, for afinancial instrument that provides to investors at least a partial hedgeagainst longevity risk exposure in at least a portion of a specificpension scheme, the reduction in the total risk capital required tosupport the financial instrument due to the diversification benefit ofthe longevity risk sources and market risk sources to which thefinancial instrument is exposed, the method comprising: constructing,using data processing apparatus, a correlation matrix of assumedcorrelation coefficients, ρ, between each pair of risk sources; andcalculating, using data processing apparatus, a diversified risk capitalrequirement, C_(total), on the basis of using the following equation:C _(total)=√{square root over (ΣC _(i) ²+Σρ_(ij) C _(i) C _(j))} whereinC_(i) is the capital requirement for risk source i and ρ_(ij) is theassumed correlation coefficient between risk sources i and j; whereinthe reduction in the total risk capital requirement due todiversification benefit is the difference between C_(total) and the sumof C_(i) across all risk sources i.
 43. A computer-implemented method asclaimed in claim 42, wherein the risk sources incorporated in thediversification benefit calculation include at least one of longevitytrend risk, mortality level risk, and process risk.
 44. A computingapparatus operable to calculate, for a financial instrument thatprovides to investors at least a partial hedge against longevity riskexposure in at least a portion of a specific pension scheme, thereduction in the total risk capital required to support the financialinstrument due to the diversification benefit of the longevity risksources and market risk sources to which the financial instrument isexposed, the apparatus comprising: a data processor; and a computerreadable media storing a plurality of computer readable instructionsthat cause the data processor to be operable to: construct a correlationmatrix of assumed correlation coefficients, ρ, between each pair of risksources; and calculate a diversified risk capital requirement,C_(total), on the basis of using the following equation:C _(total)=√{square root over (C _(i) ²+Σρ_(ij) C _(i) C _(j))} whereinC_(i) is the capital requirement for risk source i and ρ_(ij) is theassumed correlation coefficient between risk sources i and j; whereinthe reduction in the total risk capital requirement due todiversification benefit is the difference between C_(total) and the sumof C_(i) across all risk sources i.
 45. A computer-implemented method ofcalculating, for a financial instrument that provides to investors atleast a partial hedge against longevity risk exposure in at least aportion of a specific pension scheme, the total risk capital required tosupport the financial instrument taking into account the diversificationbenefit of the longevity risk sources and market risk sources to whichthe financial instrument is exposed, the method comprising: repeatedlysimulating, using a Monte Carlo method and a data processing apparatus,the projected losses on the financial instrument due to each risk sourceto produce a number of outcomes; calculating for each outcome, using adata processing apparatus, a combined projected loss as the sum of theprojected losses due to each risk source; and calculating, using a dataprocessing apparatus, the total risk capital requirement on the basis ofthe distribution of outcomes of the combined projected losses.
 46. Acomputer-implemented method as claimed in claim 43, wherein the totalrisk capital requirement is calculated as that which would result in acombined projected loss that is simulated to occur with a probabilityequal to the probability of default of a bond having an desired creditrating according to a credit rating agency's default probability ratetable for an appropriate time horizon matched to the duration of thepension scheme's liabilities.
 47. A computer-implemented method asclaimed in claim 43, wherein the total risk capital requirement iscalculated as that which would ensure the combined projected loss isless than the expected loss of a bond having a desired credit ratingaccording to a credit rating agency's idealised loss rate table for anappropriate time horizon matched to the duration of the pension scheme'sliabilities.
 48. A computer-implemented method as claimed in claim 43,wherein there is assumed in the Monte Carlo simulations and subsequentcalculations to be no correlation between the risk sources.
 49. Acomputing apparatus operable to calculate, for a financial instrumentthat provides to investors at least a partial hedge against longevityrisk exposure in at least a portion of a specific pension scheme, thetotal risk capital required to support the financial instrument takinginto account the diversification benefit of the longevity risk sourcesand market risk sources to which the financial instrument is exposed,the apparatus comprising: a data processor; and a computer readablemedia storing a plurality of computer readable instructions that causethe data processor to be operable to: repeatedly simulate, using a MonteCarlo method, the projected losses on the financial instrument due toeach risk source to produce a number of outcomes; calculate for eachoutcome a combined projected loss as the sum of the projected losses dueto each risk source; and calculate the total risk capital requirement onthe basis of the distribution of outcomes of the combined projectedlosses.
 50. A method of achieving a rating for the longevity riskexposure of a financial instrument provided to an investor, theinstrument undertaking to pay, at regular points in time over aspecified duration, sums according to a schedule of payment amountsassociated with the financial instrument, said scheduled payment amountsbeing arranged to match with the expected cash flow obligations of apension scheme to its members, said expected cash flow obligations at apoint in time being calculated at least taking into account theprojected likelihood that each pension scheme member will survive untilthat point in time, the method comprising: calculating, by performingstress tests on the expected cash flows, the amount of risk capitalrequired to be held to achieve the rating and to ensure that theexpected loss that would result from a mortality shock is lower than theexpected loss of a bond having an equivalent credit rating according toa credit rating agency's idealised loss rate table; and holding saidamount of risk capital.
 51. A method comprising: providing to an entitya financial instrument which undertakes to pay to the entity, at regularpoints in time within a specified duration, sums according to a scheduleof payment amounts associated with the financial instrument, saidscheduled payment amounts being arranged to match with expected cashflow obligations of a pension scheme to members of the pension scheme;and at a re-set point in time, resetting the schedule of payment amountssuch that the entity will receive an adjusted payment amount at ascheduled time calculated to be an aggregate of nominal cash flows to bepaid to the members of the pension scheme adjusted to take into accountactual cumulative mortality experience within the pension scheme priorto the re-set point in time; and calculating, by performing longevitystress tests on the expected cash flows using data processing apparatus,an amount of longevity risk capital to be held, and holding said amountof risk capital.
 52. A method as claimed in claim 51, further comprisingholding the required amount of capital.
 53. A method as claimed in claim51, wherein the financial instrument carries a rating from a ratingagency.
 54. A method as claimed in claim 53, wherein the financialinstrument carries a rating from at least one of Standard & Poor's,Moody's and Fitch rating agencies.
 55. A method as claimed in claim 53,wherein the longevity risk capital is raised by issuing subordinatedtranches of debt and equity capital in the form of capital notes andequity notes.
 56. A method as claimed in claim 55, wherein thesubordinated tranches of capital notes and equity notes further have anexposure to asset risk.
 57. A method as claimed in claim 55, wherein thefinancial instrument carries a rating from a rating agency and asubordinated tranche of capital is sized to have a capitalisation thatcorresponds to a junior rating from the rating agency and is positionedaccordingly in a sequential payment structure of a payment waterfall.58. A method as claimed in claim 51, wherein for a scheduled paymentpoint in time said expected cash flow obligations of the pension schemeto its members are calculated taking into account at least the projectedlikelihood that each pension scheme member will survive until that pointin time, and wherein the financial instrument carries a rating from arating agency, the rating having been achieved by performing stresstests on the expected cash flows so as to calculate an amount of riskcapital required to be held to achieve the rating and to ensure that thepayment amount obligations on the financial instrument can be met in thecase of a worst case longevity shock which is projected to occur with aprobability of no more than the default probability of a bond having anequivalent rating according to a rating agency's default probabilityrate table.
 59. A method as claimed in claim 58, wherein the ratingagency is Standard & Poors or Fitch.
 60. A method as claimed in claim58, wherein the projected likelihood that each pension scheme memberwill survive is calculated by modelling changes in the probability ofsurvival of a suitable reference population by using a statisticallongevity projection model to extrapolate, into the future, trends inthe actual mortality experience of that reference population andadjusting the mortality table associated with the reference populationto incorporate these trends, wherein the stress tests on the expectedcash flows are performed by using the statistical longevity projectionmodel to simulate the Net Present Value of the pension scheme's expectedcash flow obligations for varying future longevity outcomes; the methodincluding determining, for a plurality of future points in time and forall varying future longevity outcomes, a binding time horizon having alargest Net Present Value of the pension scheme's expected cash flowobligations that is projected to occur with a probability of no morethan the default probability of a bond having an equivalent ratingaccording to a rating agency's default probability rate table; and therisk capital requirement being the difference between said Largest NetPresent Value of the pension scheme's expected cash flow obligations anda best estimate of said Net Present Value at said binding time horizon.61. A method as claimed in claim 51, wherein for a scheduled paymentpoint in time said expected cash flow obligations of the pension schemeto its members are calculated taking into account at least the projectedlikelihood that each pension scheme member will survive until that pointin time, and wherein the financial instrument carries a rating from arating agency, the rating having been achieved by performing stresstests on the expected cash flows to calculate the amount of risk capitalrequired to be held to achieve the rating and to ensure that theexpected loss that would result from a mortality shock is lower than theexpected loss of a bond having an equivalent credit rating according toa credit rating agency's idealised loss rate table.
 62. A method asclaimed in claim 61, wherein the credit rating agency is Moody's.
 63. Amethod as claimed in claim 61, wherein the projected likelihood thateach pension scheme member will survive is calculated by modellingchanges in the probability of survival of a suitable referencepopulation by using a statistical longevity projection model toextrapolate, into the future, trends in the actual mortality experienceof that reference population and adjusting the mortality tableassociated with the reference population to incorporate these trends,and wherein the stress tests on the expected cash flows are performed byusing the statistical longevity projection model to simulate a NetPresent Value of the pension scheme's expected cash flow obligations forvarying future longevity outcomes; the method including determining, fora plurality of future time points and for all varying future longevityoutcomes, a binding time horizon having the largest Net Present Value ofthe pension scheme's expected cash flow obligations that results in anexpected loss of a bond having an equivalent rating according to arating agency's idealised loss rate table; the risk capital requirementbeing the difference between said largest Net Present Value of thepension scheme's expected cash flow obligations and a best estimate ofsaid Net Present Value at said binding time horizon.
 64. A method asclaimed in claim 51, wherein the calculation of the amount of riskcapital required to be held to achieve the rating specified by therating agency further comprises: characterising, by performing abootstrapping analysis on a reference population, an error distributionfor the mortality projections produced by a statistical mortalityprojection model, the error distribution being associated with a size ofthe population of the pension scheme; and determining, by applying saiderror distribution to the Net Present Value of the expected cash flows,the amount of risk capital to be held to ensure that: the paymentamounts on the financial instrument can be met in the case of a sampleerror in the mortality projections which is projected to occur with aprobability of no more than the default probability of a bond having anequivalent rating according to the rating agency's default probabilityrate table; or the expected loss that would result from a sample errorin the mortality projections is lower than the expected loss of a bondhaving an equivalent rating according to the rating agency's idealisedloss rate table.
 65. A method as claimed in claim 64, wherein saidbootstrapping analysis comprises: calculating, for N random samples ofmembers of the reference population of the same size as the populationof the pension scheme, the mortality rate projected by the statisticalmortality projection model for that random sample for a period of time;comparing each of said mortality rate projections with the actualmortality rate for that sample of the reference population and for thatperiod of time to determine errors in the mortality projections; andcharacterising the distribution of the errors in the mortalityprojections.
 66. A computer-implemented method of calculating, inrelation to a financial instrument that pays to an investor a cash flowaccording to a payment schedule arranged to match with at least part ofthe cash flow obligations of a pension scheme to at least a portion ofits members such that the financial instrument provides to the investorat least a partial hedge against longevity risk exposure in said pensionscheme, an amount of risk capital required to support longevity trendrisk exposure in the financial instrument and achieve a specific ratingfor the financial instrument, the method comprising: calculating, usingdata processing apparatus, the expected cash flow of the financialinstrument at a time horizon at least taking into account the projectedlikelihood that each pension scheme member will survive until that timehorizon, calculating, using data processing apparatus to perform stresstests on the expected cash flows of the financial instrument, an amountof risk capital required to be held to achieve the rating and to ensurethat the payment amount obligations on the financial instrument can bemet in the case of a worst case longevity shock which is projected tooccur with a probability of no more than the default probability of abond having an equivalent rating according to a rating agency's defaultprobability rate table; and holding at least said amount of riskcapital.
 67. A method of quantifying the longevity risk exposure of afinancial instrument provided to an investor, the instrument undertakingto pay, at points in time over a specified duration, sums according to aschedule of payment amounts associated with the financial instrument,said scheduled payment amounts being arranged to match with the expectedcash flow obligations of a pension scheme to its members, said expectedcash flow obligations at a point in time being calculated at leasttaking into account the projected likelihood that each pension schememember will survive until that point in time; the method comprising:calculating a present value of the financial instrument from theexpected cash flows making up the payment schedule; and calculating, byperforming stress tests on the expected cash flows, the change in thepresent value of the financial instrument that occurs due to a longevityshock that is projected to occur with the specified probability, thechange in present value representing the risk exposure of the financialinstrument to the longevity shock having that probability.
 68. A methodof quantifying the longevity risk exposure of an asset or a liability toa longevity shock that is projected to occur with a specifiedprobability, the asset or liability having cash flows of sums ofaccounts receivable and accounts payable at regular points in time overa specified duration, said sums being at least a function of the actualmortality experience of a group of creditors or debtors, comprising:calculating, for each point in time, the expected cash flows at thatpoint in time at least taking into account the projected likelihood thateach creditor or debtor will survive until that point in time;calculating a present value of the asset or liability from the expectedcash flows; and calculating, by performing stress tests on the expectedcash flows, the change in the present value of the asset or liabilitythat occurs due to a longevity shock that is projected to occur with thespecified probability, the change in present value representing the riskexposure of the asset or liability to the longevity shock having thatprobability.
 69. A method as claimed in claim 68, wherein the asset orliability is selected from the group comprising: a defined benefitpension scheme; a defined contribution pension scheme; one or moreequity release mortgages; one or more reverse mortgages; and a financialinstrument that is arranged to transfer the longevity risk exposure ofany of these assets and liabilities to the capital markets.